You've made a new custom character and you want to powerlevel it or you're starting out and don't really want to deal with the leveling aspect, there are a few tricks you can use in Dragonball Z Xenoverse to level up quickly.  

Let's talk a bit about changing your appearance / attributes before you want to make a new character to grind up. 

Changing Appearance / Attributes (Stats) 

Misspend your points or hate how your character looks? Well, remake it, but if you want a challenge you can collect the 7 Dragonballs, then go to Shenron who will grant you the wish to be Drop-Dead Gorgeous (appearance) or a Second Chance at Life (attribute reset 

Parallel quest #15 has a chance to disperse a Dragonball, it's recommended to farm it if you want to level an alt and want Dragonballs. 

Otherwise, there is currently no other way to reset your attributes / apperance and you can not change your name, race or gender. To do that you have to remake, as there is no respec for that.

I really don't suggest you do it though, you have to farm all of the balls and use them and it's kind of a waste when you can just remake and grind the balls on a new character leveling it up and then cash them in for an unlock.

Leveling First Character 

Oh boy this one is kind of tough, because without parallel missions, you literally only have the story to complete. Which is the only way to level up quickly, since the story actually gives a lot of XP per mission and has so very many missions to complete. 

If it's your first time through, just... do the story missions. I hate that advice, because it's so cheap, it's not a good solution, but it's the only solution. It's important because it'll let you level up other characters faster as well, since you'll unlock skills and missions and the such. 

Leveling Up New Characters 

Want a different race, gender, or an easy to change look / attributes? Well new character is the way to go! However, you'll have to level up, which is a bit lame and kind of hard to do. What I've done so far is do really high level parallel quests, like 40-50, and even if you fail them (which you will), you'll gain the first 20 or so levels rather quickly. 

Otherwise, you can grind parallel quests day in and day out to level, there are some better than others, depending on your build. There is one I would like to note, any level one character can complete it, all you have to do is unlock 53 parallel quests and do parallel quest 53 which only requires you to return 3 Dragonballs to get a clear. Your sidekicks will keep Broly busy while you snag them. 

For me though, by the time you're that far in the game, you'll probably realize that using the premade characters is easier unless you have a specific build in mind, however, if you want to level up fast and do not have quest 53 unlocked, I suggest #8 which is similar to #53 but wayyy less score, #15 which has the added bonus of being a Dragonball farming spot, #23 which is easy, #28 which is similar to #53, #31 which is easy (to me at least), and really any quest involving a one on one fight. 

#15 is noteworthy as you can get a Dragonball by defeating the Time Patroller, farming this to level up is smart because you get... well Dragonballs which are good things and help you unlock characters and do stat resets. 

A note that #53 gives something like 15k score, the rest of these grant somewhere between 2 and 8k, up to 10 to 20 if you can do the ultimate finish. 

Grinding shouldn't be too painful, since you'll mostly get to play matches over and over again, which is sort of the point of the game.