You need a 26" MTB wheelset to match your brakes and drive train; ie, have the same mounting width, rim diameter/width and accepting your freewheel/cassette. I don't think you can use 1.0" tires, but I've been out of cycling a long time and there are many new things I'm catching up on. In other words, it probably would be best to get a set of wheels that you can use with 2.0" MTB tires or 1.5" general use tires with a smooth ride, like the 'Contact' by Continental, and there are many others at all price ranges. For the tube, you get the size that includes your tire size on the box. Also make sure the valve type matches the hole in the rim (Schrader/big hole, Presta/small hole) but they also make a plastic insert to convert a Schrader hole to a Presta. For changing tire sizes, you'll need a tube for each size.
Before buying an expensive wheelset, at least have your wheels looked at for cleaning up, re-truing and overhauling the hubs. True, if the rims are too beat up, it may be a show-stopper, but you can use several different 26" tires on your MTB rims.