Recently, I posted pychess.org as a great place to play many different chess variants. But I have also found some other great sites, as recently as today:
pychess.org
lichess.org for live chess
lidraughts.org for live checkers and variants
lishogi.org for full board shogi (no variants yet, beta version)
online-go.com is the go to site for go (OGS)
xiangqi.com for Chinese Chess
For more turn based games similar to littlegolem, there is also yourturnmyturn.com
Anyone know any other good sites to play strategy games?
BoardGameArena has quite a number of pure strategy games now (50+), and it fully supports turn-based play, as well as real-time ones. There’s a list of all the relevant games in this group here:
PlayOK is great for live card and abstract board games. It runs in HTML5 and has a fairly large user base for games like Xiangqi, Makruk, Backgammon, and Canasta, particularly if you're on at certain times (it's almost noon in Thailand right now and Makruk has 900+ players live)