No, you have about 4 specials, each one being one of the four face buttons on the controller (you have to be holding R2 beforehand to get the menu of your specials up I believe). Those four special moves range anywhere from Kamehameha to a counter to something that powers you up (like a Great Saiyman's pose) to Ki volley moves. You can make those 4 specials whatever you want, so you could have Kamehameha and Galick Gun on your custom character if you wanted but the normal characters like Goku and Vegeta have pre-set move sets.
However, those characters have tons of different alts so when you select an alternate costume not only is the costume different but their move set also differs. So, if you pick Goku with his shirt completely ripped off he has 3x Kaioken in his move set instead of whatever he might have had before. It's not just one move that changes, of course. This is how characters get their transformations as well. So, if you pick Halo Goku he has Super Saiyan 2 as his transformation.
The way transformations work in this game is that they count as an ultimate slot. There's two ultimate slots per character and you don't need a previous transformation to transform into the next. For example, you don't need SSJ to transform into SSJ2, you only need to equip SSJ2 to transform into SSJ2. So, for your custom character you can either have two ultimates or one ultimate and one transformation (the choice between SSJ or SSJ2, basically). The original characters have transformations based on their alternate costume chosen, you cannot change the original character's move sets.
But these alternate costumes, while they may seem redundant and superfluous (why not just let us change the moves) they actually have different stats from each other. For example, the ripped shirt Goku from the Saiyan saga has a lot more stamina than the default Goku. This is so he can use Kaioken for a longer period of time. He has a slightly smaller Ki bar in this outfit as well. And, of course, the other stats differ but those are the main two that are immediately noticeable upon first glance.
As a result, the alternate costumes are like entirely new characters except the character does not get any new combos or melee strings, they keep the same strings as their default costume. Therefore, they don't actually count as new characters.
Anyway, I hope that made sense. To sum up, if you're a custom character you can customize their move set however you want but if you're an original character you can't and you have to pick a different alt. costume to use whatever form and moves you want. If you want SSJ3 Goku there's two alts that have that each with different moves and stats. If you want SSJ Goku there's also a pair of alts that have it each with different moves and stats from the original Goku.
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