2 winters ago I was in the same boat - 2yr old and an infant. I tried to get out as often as I could. A friend and I started walking the Palisades mall. Another friend and I would meet up at Ikea and let the toddlers play in the kids' area and sometimes "try out" different couches/chairs.
In terms of indoor activities:
I'm a big fan of "swimming" in the bathtub. It gives them exercise and it's a fun way to break up the day. You can give your 2 yr. old tub crayons/paint or just blow bubbles into the tub.
Stickers, watercoloring, and play doh were favorites for that age.
Cook simple things like cornbread where they can pour in the water and help stir.
We got a blow up pool and filled it with ball-pit balls (you can get a lot really cheap) and let our 2 yr old go nuts. She wore herself out and it gave her a lot to do - every time she knocked all the balls out - she had to stop and gather them all and then "shoot" them into the pool.
Making forts with blankets and furniture was fun.
Play dress up with my old scarves - we even used one like a mini parachute where we shook it to music, etc.
Music was always a good way to transition too - have a dance party or just play soft music if your child is getting antsy with whatever activity you want them to do.
I got my 2 yr old involved in a lot of the daily chores the best way I could - she handed me clean laundry to fold, she used her little broom while I swept the house, she looked under the couch before I vacuumed.
The thing that kept me sane during the winter was to create almost like a school schedule for us at home. For the first hour we were in the livingroom, then we spent an hour in her bedroom, then we spent an hour in the basement, etc. We both got bored quicker when we were spending too much time in one spot or randomly went through the house. If you can make yourself a list of things to do during the day and just go through them, it's a lot less exhausting then constantly trying to think of something on the fly.
Good luck!