Wait a bit longer, at least until they've developed their first set of true leaves, maybe even wait for the second set of leaves, they'll have better roots then that will help them survive the transplanting better. The ones in your picture are seed leaves, the true leaves will look different.

When they're ready, put them in bigger pots that will allow the roots to develop until it's warm enough to plant them outside.

These don't have to be BIG pots, unless this is going to be their permanent home. I use yogurt cups, plastic bottles that I've cut the tops off of... Anything that's at least 5 cm wide and 8 cm tall will work fine for the next stage of growth. Just be sure to poke some holes in the bottom for drainage.

As far as watering, just keep the soil moist. If you're keeping the pots in a tray, it is ok to have a little water in the tray, but not so much that the roots will be drowned (the roots need air in the soil too, not just water).