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Your best bet is to join a casual guild that does teaching runs for normal mode trials. The normal modes aren't that difficult, but there are some mechanics you need to learn. Aetherian Archive ("AA") and Hel Ra Citadel ("HRC") are the easiest. Sanctum Ophidia ("SO") is a bit harder, and Maw of Lorkhaj ("MoL") is the hardest. Be aware that you may not end up seeing all the mechanics in normal mode, as things tend to die too fast.
Before you start, you will want to research some builds for your class as well. Look for end game builds, like the ones here : http://alcasthq.com/
Copy down the list of gear that you will need, and work on obtaining that. Be aware though, those builds may change when the Housing patch hits, as some things are being changed that affect some sets, skills, and weapons. Get the proper skills leveled, and work on the rotation. Once you have that down, then you are ready to try normal trials.
Once you have gotten your DPS to about 30k, have the right gear, and have become familiar with the mechanics in the normal modes, then you can start looking for a group that does veteran versions. I would recommend you look up boss fights for the vet versions, as you won't see all the mechanics by doing normal - and vet is A LOT harder, especially vMoL.
You may also want to grab some add ons if you are on PC. Here are some I use:
CombatMetrics: a DPS meter, plus it has a lot of other useful info
RaidNotifier: flashes warnings about incoming attacks and debuffs that are on you - I highly recommend this one for vMoL
PapaCrown: makes the crown over the leaders head bigger, so it is easier to see
ActionDurationReminder: lets you know when your abilities with durations run out - this is good for perfecting your rotation
There are many more that are useful, so maybe some others can give more suggestions.
Scroll all the way to the bottom until you find "Guild Recruitment". Look for your platform/server. Read through the offerings and see if one might fit your needs. You can be in 5 guilds, so try some different ones out until you find one that fits you.
Random Trials seems to start up in craglorn and happen often throughout the weekend... more so in the range 5 to 8 pm est, but I would start looking around noon.
Reaper's march has dlc trials starting there.
mournhold also seem the be more popular capital for such events, followed by elden root ... and wayrest
Type in zone chat
Dps LFG for trials
300 dps LFG for trials
I suggest you attempt a normal raid at cp 120... it's possible... lower... but amount of time spent
I suggest you attempt a veteran trail at CP 300 and have 20k dps min