ah the priest. such and excellent and versatile class to play!
well first off don't worry about arenas yet. you wont be able to get into one that is worthwhile anyway until you hit level 80. at that point you will have to pay for a team charter (which you get to name) and then you'll have to find other players to play with (you can have a 2's team, 3's and 5's) try to find people you know will be playing at the same time as you as arena points allow you to purchase pvp gear etc.
as for what spec you are, check out your talent tree. the tree tabs are set up like this: discipline holy shadow.
how many points do you have in what tree? if you have most of your points in a specific tree then that is what spec you are. (my priests main spec is holy, her off spec is shadow)
try to focus all your points in one tree as your abilities are specific to your tree. painsuppressionn, grace and penance are talents in the disc tree. discpriestss are ideal main tank healers, as well as a great pvp spec. Holy is an excellent raid healing spec, with talents such as spirit of redemption, circle of healing and guardian spirit. Shadow is strictly a damage tree, so the only points you want in that tree for leveling (if you're leveling discipline) is spirit tap, which helps you to regen mana at a faster rate.
losing in duels is not a big deal. you just have to remember that fear is your best friend as a priest. what you need to learn to do is shield yourself pretty much constantly. shield before the duel starts to begin the cooldown. dot the other player with shadow word pain and devouring plague. (now this part depends on your spec) cast mind flay if you're shadow, penance if you're disc and smite if you're holy. when the player gets close enough to you cast your fear and re-shield yourself. if you're shadow cast mind blast and then mind flay, if you're disc cast smite and penance, if you're holy cast holy fire. this is a long casting spell but it does a large amount of burst damage as well as putting up a dot. refresh your dots if they're up. remember you have your little shadowfiend that you can use to replenish mana. (actually im not sure what level you get that pet at, im 80 on my priest so it's hard to remember =P) with your shield up you can cast SW:death without causing yourself damage because you're shielded.
it's hard to duel when you're healing spec.
you should post an armory link to your toon. it's easier to tell you if you're gear is the problem or if it's your dueling abilities >.<
good luck! see you in azeroth :)