Well, the laws of the seder stipulate that four cups are to be drunk as part of the seder itself. So, you're supposed to drink four cups. A couple of points to keep in mind:
1) When drinking the four cups, it is not necessary to drain the cup each time - however, one is supposed to drink "rov kos," that is, the majority of the cup.
2) Grape juice MIGHT be acceptable in lieu of wine.
3) A lot of wines that are made for Jewish ceremonial use are of much lower alcohol content than normal wine. If the bottle says something like "6% alcohol," then one glass of that stuff is the equivalent of like, half a cup of regular wine.
The second point is the subject of debate among poskim (Jewish legal authorities). That being the case, you should consult your LOR (Local Orthodox Rabbi). Some might tell you that it is preferable to use grape juice, particularly if you don't like the taste of wine.
For sure there is no reason for you to get drunk at the seder, and it wouldn't be appropriate (Purim is when you want to get good and shikkur). I'm not qualified to tell you what you can or can't do from the perspective of Jewish Law, so consult with your rabbi about any or all of the following:
1) The permissability of using juice instead of wine.
2) ...or diluting wine with juice
3) How much of the cup you actually need to drink.
4) How big the cup has to be. (A shot glass is too small, but you don't need to pour a huge cup or anything.)
Of course, this is if they serve you wine in the first place.
I hope you find this information helpful.
There may or may not be dancing at your seder... the last time I went to one where there was dancing, it was at the very end, probably to celebrate that we'd survived the whole thing as much as to celebrate the holiday itself. ;)
PS: Just a minor correction to one of the other answers - a girl becomes a bas mitzvah (an adult for the purposes of Jewish Law) at age 12. A boy becomes a bar mitzvah at age 13. From what I can tell, it's generally understood that girls mature faster than boys.