You said you recently moved in and that's when you started getting the bites. Fleas live outside in the grass and if your place has a carpet, and the previous residents had pets, the eggs can live in the crevices of the carpet nap.
Also, if it's bed bugs - I've never had any experience with them but have seen documentaries about them. They're very difficult to kill, they can hide in any crevice on the floor, the walls, carpets, etc. From what I've read, the only way to kill bedbugs is to call in a professional exterminator and have them treat your place with heat. The little critters are practically indestructible to any chemical treatments. Once you get bed bugs, you have to get rid of your bed, couches, essentially anything that has stuffing in it. That includes carpets also. These little unwanted tenants may have been in your place before you moved in.
You can however check for them. Check any creases on your mattress, they
thrive seemingly unnoticed and come out to feast. You can also check the edges of your carpeting and if your bed base is wood, they'll hide there too.
I hope it's not bed bugs, they are a nightmare from what I've read. I've always been "paranoid" about it because I always had little ones and made sure we didn't share our living environment with them critters LOL.
If it's other insects try using Avon's Skinsosoft. They have a whole line of insect repellents.
Have you stayed in any hotel/motel recently? That's another place your clothes may have brought them in.