The top one with the little white flowers is a Fukien tea, aka Carmona. The bottom one is a ginseng ficus. Both of them are tropicals that can handle being indoors in winter, but like most trees, will benefit from being outdoors in spring and summer as long as temps are reliably above 50 or so.
Bigger pot is kind of up to you and what you want to do with them. Bonsai is usually about keeping trees in miniaturized form in small pots. If you want to let your trees get bigger, and particularly if you want them to develop thicker trunks, then put them in bigger pots. But more important is what is IN the pots - what kind of planting medium they are in. I would lift the moss off the top of the soil and see what it looks like under there. If it is like potting soil, soft and dark and peat mossy, you should change that for real bonsai soil, which is very mineral and gritty.
Get a couple wooden chopsticks. Put them an inch or so down into the soil and leave them for awhile (at least a few minutes, up to all the time). Pull them out and check. If the chopstick is dark and damp, the soil is damp and does not need watering. If the chopstick is dry, then the soil is dry and should be watered. Don't water on a schedule, water when they need it, which will vary depending on heat, humidity, sun, etc.
Other wise, start doing a little reading on bonsai.
Here's some inspirational images for you of what your bonsais could look like a few years from now, if you get really hooked on the art of bonsai:
https://www.artofbonsai.org/art-of-bonsa ... phylla.jpg
https://img708.imageshack.us/img708/3974 ... ginsen.jpg