Dragon Ball is the story of Son Goku (Kakarott), adopted and trained by his "grandfather" ever since he was a baby, Goku strives as a martial artist. Dragon Ball is a story which starts out as a light hearted comedy martial arts manga, all beggining when Bulma meets Goku in the mountains. Bulma and Goku team up to find the dragon balls and go on an adventure to do so. The story then embarks on story arcs/sagas about Goku training under master Roshi to fight in the world tournament, Goku taking on the criminal army Red Ribbon in the search of his precious memento the 4 star dragon ball, Goku eventually entering another tournament where he fightsTien Shin Han, the disciple of his master rival. The story does not become fully serious till the introduction of Piccolo, whom in his Daimaoh incarnation takes Krillin's life and tries to become the planets ruler. King Piccolo is eventually defeated by Goku, but just before his death he spits out an egg which lands too far away to distinguish, the egg carrying his reincarnation which will eventually try to fight Goku again for revenge. Goku goes to train with Kami, earths god, in order to prepare for the upcoming battle, 3 years later, Piccolo jr. and Goku are ready and clash in the 23rd Budokai tournament, Goku eventually claiming the win once again and also reuniting with Milk who would be his wife.
Part 2 (DBZ):
Over 4 years have passed since the defeat of Piccolo, Goku is now married and has a son named Son Gohan (in honor to his "grandfather), however peace is disturbed as a mysterious warrior arrives on earth. The mysterious warrior is Raditz, he is Goku's brother and a warrior of the extreterrestrial race known as the saiyans (saiya-jins). He proceeds to ask Goku for help in order to conquer the planet and also another planet they have in sight after earth, however Goku refuses and in order to force Goku to accept his demands he takes Gohan hostage. Goku and Piccolo team up to take on Raditz who had proved far too much for any one of them on there own, eventually defeating him in exchange of Goku's life only to learn there are 2 more saiyans in the way both being much stronger than Raditz himself. After this the story tells on how Goku and the other warriors continue training in there quest to be the strongest and protect the earth, the first menace after Raditz being Nappa and Vegeta the prince of saiyans. After this the story continus to unfold telling the legend of the super saiyan (super saiya-jin) and the face-off against the series most famous and noteworthy villains: Frieza, Cell and Buu. Part 2 also has a heavy concentration on Gohan, who would be somewhat like the stories second protagonist.
Part 3 (Post-Buu arc/Super):
Dragon Ball is being further continued even years later after the original series ended in what could be called a "part 3" to the story. After the defeat of Buu there are 3 canon events so far which take place in the storyline.First is Yo! Son Goku and his friends return, a special chapter released due to the anniversary of the serie. 2 years after Buu has been defeated Mr. Satan opens a new hotel and has a party, during this a space pod crashes on earth with Tarble, the brother of Vegeta. Tarble comes explaining that 2 of Frieza's elite soldiers rose in power after Frieza's defeat and became as powerful as Frieza himself, and asks for the help of the Z warriors since they are chasing him down. This chapter is mostly comedic relief.
A year after this happens Battle of Gods, the only canon movie so far written by Akira Toriyama himself. In the events of this movie, Bills/Beers the God of destruction awakens and goes in the search of the alledged "Super Saiyan God" who he would supposedly have a grand battle with according to a propechy. No one can stand up to Bills/Beers overwhelming power till they discover the method to reach the form of Super Saiyan God, and Goku faces off once more against Bills.
Furthermore a new movie is being released called "Fukkatsu no F" in which Bills/Beerus and Whis are making a return and in which Frieza is being revived and powered up. The movie is set to be premiered in April, 2015.
A few years later after this events Buu reincarnates as Uub and Goku finds him to train him, which is the officially final chapter of the serie so far. A timeline by Akira Toriyama for Dragon Ball Online was also released in which the death of Goku and Vegeta is noted, but its canonicity is unknown (but it is endorsed by Akira Toriyama and made by himself, so it should at least be valid for feats)
The Dragon Ball has a well defined canonicity, the manga being the main canon, along with Yo! Son Goku and his friends return, Battle of Gods and Fukkatsu no F. Daizenshuu's would be a second line of canon, being done by Bird studios, Akira Toriyama's private studios. Dragon Ball Online and the anime would be more or less an even lower grade of canon and should not be used if it contradicts the manga, preferibly not used at all. Bardock - The Father of Goku (a tv special) is apparently canon as well as it is referenced in Toriyama's own works. The rest of the movies and GT are non-canon.
Despite being quite powerful, the power of the verse is a highly contributed subject, likely due to most people using the anime and not the actual manga. The manga itself is also somewhat inconsistant and can not be taken overly serious until the Piccolo plot appears, however some level of consistancy can still be found (mostly the scaling), example:
-King Piccolo is a city+ buster
-Raditz and Nappa are moon+ busters
-(Scouter/Saiyan saga) Vegeta is a planet buster
-Base form Frieza is a casual planet buster
-Somewhere around Frieza or above we reach multi-planet to sun busting
-Super Perfect Cell is a solar system buster
and so on.
Part 1 tops around moon busters with PJR and Son Goku, reaching up to mach 50+ speeds and at least hundreds of thousands of tons of physical strength. Part 2 starts around moon level, but quickly becomes planet lvl, and reaches up to supernova lvl in the higher parts of the serie (based on the DBO timeline written by himself and Cell's solar sytem busting claim). Besides there destructive capacity they also reach insane striking strength and easily faster than light speeds (going by the actual series context). The series strongest character, Vegito, is in a class of his own effortlessly owning the second strongest character shown despite only using a portion of his power. That was until Battle of Gods happened, in which the higher gods dimension of power was introduced, in which the higher ranking gods use there own special type of energy which is superior to ki, an energy called "God aura". All beings in this level are in a superior dimension of power to even Vegito. Battle of Gods also introduced the fact that there are 12 different universes in the DB-verse, each with its own God of destruction.
High tier DBZ reaches characters that are able to give the impetu for life up to such things as planets and stars, and guys like Bills/Beers the God of destruction, who can dissapear the solar system in an instant. Furthermore, Whis (the attendant and martial arts master of Bills/Beers) is implied to be considerably stronger than even the God of destruction himself.
Clear outliers regarding the series power:
-Master Roshi destroying the moon in DB pt.1
-Goku struggling to shadow box with only 40 tons in Buu saga
-A good deal of showings involving travel speed
Something else quite debated about Dragon Ball is the size of its universe. The therm 4 galaxies is used for the 4 quadrants of the universe, but it is never made explicitly clear if they are solely 4 galaxies or something more beyond mere literal galaxies.
For part 3 (Dragon Ball Super) the verse has finally confirmed speeds that are far above FTL and has solid universal to low multiversal beings.
Gohan (Cell Games, Buu saga)
Upper mid tiers:
Goku (Androids saga, Cell saga)
Vegeta (Androids saga, Cell saga)
Future Trunks (Cell saga)
Goku (Frieza saga)
Vegeta (Frieza saga)
Tien Shin Han (Androids-Cell saga and above)
Krillin (Androids-Cell saga and above)
Yamcha (Androids-Cell saga and above)
Calcs, Feats, etc.
There is plenty evidence regarding DBZ being stronger than usually given credit for due to its lack of feats, here such evidence will be listed.
Potential FTL movement as far back since during Frieza vs Goku: https://www.facebook.com/groups/454033638040208/permalink/610295589080678/
Casual, spammable ki blasts in the Buu saga are planet busting: http://ex1.unixmanga.net/onlinereading/?image=Dragon+Ball+Complete%2FDragonBall+Vol+042%2FDb42ch06%2F420605.gif&server=nas.html
Cell's solar system busting "hyperbole" being confirmed in a guide made to analyze DBZ up to the Cell saga "Page 147
Super Saiyan Grade Five (Strongest Form) An aura with sparks like flashes of lightning and an upright, combative hairstyle are the distinguishing features of this, the strongest Saiyan warrior! One's personality also becomes aggressive; even the ordinarily gentle Gohan started to enjoy battle! Its power is enough to push back even Cell's energy bullet, which had enough force to blow away the Solar System!": http://www.kanzenshuu.com/forum/viewtopic.php?f=7&t=22411
Supernova level explosions in Dragon Ball Online timeline made by Akira Toriyama by an old Goku and Vegeta ear there deathbeds and years before Battle of Gods and any of its concepts where even planned: AGE 801''Son Goku vanishes.
--Knowing his end is near, Goku leaves Earth with Vegeta so that the two can finally settle their score. Several years later, [astronomers?] note the explosion of a supernova. Possibly this was an aftereffect of Son Goku and Vegeta’s battle.: http://www.gamefaqs.com/boards/2000113-/61998201
Bills/Beerus being stated of being capable of destroying the Kaiohshin Realm which is aproximately 1/10th of the universes size: http://www.kanzenshuu.com/translations/saikyo-jump-june-2014-issue-twel-buu-mysteries/
Dragon Ball 7th universe map showcasing the size of the Kaiohshin Realm: http://i164.photobucket.com/albums/u40/hellmasker-d/MapofAnoyo.jpg
And a Daizenshuu explanation confirming it "“IV. The Kaioshin Realm
界王神界/kaioushin-kai Translation: Kaioshin World, Kaioshin Realm Other Names: Planet of the Kai (Funi), Kaioshin’s faraway planet (Viz)
The Kaioshin Realm is, well, the Realm of the Kaioshin. It is a special realm completely separate from the macrocosm that the afterlife, universe, and Demon Realm are all enclosed within. It's made up of a giant crystalline sphere, about 1/10th the size of the macrocosm, and it revolves around the macrocosm like a moon.": http://www.kanzenshuu.com/forum/viewtopic.php?f=7&t=11596