If you want to go to icc the right way (killing the lich etc), there are 2 ways to do so: either join a guild, or, very rarely, there are semi-guild runs (again, depends on the rl). Contrary to anything players try/announce on global (today saw a pug for 5.4k gs aiming for sindy xd), the groups there simply do not go much further than bpc - lk runs usually reach vdw and disband. So, don't depend so much on them xd
If you are more for the group experience, in deed, the best things to do are - guild - or, you prefer to be not associated with any guild, weekly raid quests, eof from rdf (#grind), and alt runs on icc 10, later 25 - most players, sadly, organise these only for the boe items & primordials, and really don't care for the rest of the raid (one of the reason actually to disband right after bql), however you will see & feel the setting around you, have some raid time, etc. Do keep in mind however, that you will NOT get a lk achievement, nor even see him, this way!
Also, don't worry too much for being full pve - in fact, there are two slots, that are acceptable per class as the items going there are rather rare to come across, and they are.. fine - ie, they can be a cloak and a wrist. Ofc, try your best to change them asap! Try to get the cheaper boes for your class as well - wodin, the cloth shoulders, the special boss herding gun, the 75g holydin abyss crystal; a cloth item on a caster dudu is perfectly fine as well. Look for the icc loot tables to see which slots you can forget about getting from raids for the moment - which item drops from each boss? Like, caster trinkets are from toc 25 (245, but is actually quite a good one!), icc 25 - rotface (mages, destro, etc) & sindy (dotters like shadow priests); the icc 10 armory gives a good 251 (264 hc) as well; 1 one of assassination rogue best trinkets is actually for eof and it can be your first item to buy with them.
And, if you want, try to make your raids as well :) Thing here is, though, that, quite often, there are other players that put a slightly higher gs req for their raids once you start your own - because, see, if anyone has 5.5 gs, where would he enter - in a 5.5 or a 5.4 raid? Still, happily, not everyone is such a troll - try several times, and you will make it! Lead well, and you will actually enjoy the raid more than if you just gather items like in a job, and be looking for a group... and, you will make them the way you actually like; given that you are a girl, i kind of think you will do it way better than the majority of the population here!
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i really plan one day, as i have a few icc hc items - not part of a set however - to boost my gs to like 6.2, and see what would a random rl say about that xD actually, no, pff i already know the answer...
Oh and one more important thing.
The moment you enter Icc, you will find yourself in the world of Recount - ignore this. Simply because it shows nothing about what you actually do - the weapon you have, sadly, matters way too much; and the best ones drop from the lich king (or are bought from the shop, however this is more or less cheating - if you want to be op, it would be best if you do it yourself and then feel the pleasure of that; coins have a way better use for scrolls, mounts, and tmog items...). I remember been on a semi-guild run, where all but one person (the thinking one in fact) were with 284 weaps, yet were doing so random things that you would totally youtube them - not even worth remembering; a very direct example - sharing recount on bql. Just set the word Recount in your chat filter as a bad one, and live freely :d Oh, and in the end, i was second on dps, with a 251 weap...
Lastly, let me quote Prot... remember to have fun! :) If you cant get in icc, try to find the cool things in wow - lead lower tier raids which many new players want to go to, however no one is making groups for reasons (primarily because they have not seen anyone to); for me, this is actually levelling (i know i will be hated for this xd); do your weekly q raids & some quick voa & icc 10 3-boss runs any time you want, explore and stuff... anything you feel like you *have to*, is a reason to bless it be with Sargeras. Not our problem.