What's the learning curve? I've been able to pick up some things about the gameplay from watching it, but I won't dare claim I know much of anything. The item system in particular seemed interesting, but complex.
Roughly speaking, there are three difficulty levels: beginner bots (AI), intermediate bots, and PvP. You can learn to hold your own against beginner bots pretty easily--maybe in a single evening, certainly in a week.
The item system is complex, but you don't have to understand it in depth to play, even at intermediate. Finding a good item build for any given champion is as easy as Google.
What's the overall network environment like? Lots of laggers and latency, or pretty solid?
Pretty solid. I have a very slow computer, so I get lag-killed once ever few matches, but my friends with real gaming machines rarely have any trouble.
I understand it's a free to play game. Now, correct me if I'm wrong, but you can purchase other characters. Are these characters available via unlock through other means, or purchase only? Does this unbalance the game in any way?
Any character in the game can be unlocked using IP, which are earned simply by playing. Real money is merely a shortcut. And there's a new set of free-to-try-this-week champions every week, so with patience you can always try a champion before you decide whether to spend IP on it.
I know with any gaming community it's a tossed bag, but how friendly and open is the community towards new players? Would I be alright to go in solo, or should I find some people to help me out?
I've been impressed. Many players are actively friendly. In bot games, most players will simply ignore and work around incompetent teammates rather than flame. I suspect that PvP would be less forgiving, but I haven't tried PvP with strangers yet. Even against bots, I would recommend starting with at least one friend (as I did) to show you the ropes. Feel free to PM me if you'd like my username.
ETA: I just remembered that unapproved users can't access PMs. But assuming you become approved, feel free to PM me :) .