Click on the gear icon in the text UI box, and set up the tabs you want, generally you want to turn off loot rolls and zone chat and such for partying. The default chat tabs are pretty messy, but once you spend 5 minutes setting stuff up you can get it pretty noise-free.
As for grouping, yeah, that's a bit sucky, no idea why everyone hasn't just copied DDO's LFM system (in a lot of moments I felt it was quite obvious Neverwinter devs are familiar with DDO, so no ide why they haven't copied it) when it's still probably the most simple and intuitive one.
Anyway, you queue up (K key) for skirmishes, dungeons and PvP (you can queue up while solo, or in group, as long as you don't have a full group). You don't have to have the dungeon/skirmish in your log to queue up for it. Depending on your class, time of day, other factors, it can take anywhere from 5 minutes to several hours to get a group - but the thing is, you can do other stuff while queued up. So generally, it's best to queue up for stuff in your level range when you log in, then do the explorer area quests, and accept group if one pops up.
As for grouping in explorer zones (no idea what they're actually called, I just call them that

) and instanced quests (the ones you can do solo and can't queue up for), you'll need to do that manually (since AFAIK they're designed for 1 player + companion), which is very painful - you either have to spam zone chat (and a lot of people have zone chat turned off) or invite someone you notice is running around the same parts of the map as you

As far as I've noticed, the best way to group for zones is to just try out the chat once when you just start and see if there's anyone else in the zone who needs all the quests done. Wouldn't know really, I soloed all the zones and instances, other than one time some guy invited me, killed stuff with me for 10 minutes then went afk and I dropped group