Replacing 2 old light switches - what kind of switches do I need?
You have 2 each single pole switches the black wire that connects to both switches is the hot wire (turn the breaker off before replacing) any standard single pole switch will work, you can get simple toggles like you have or the fancier decora but a new cover plate will be needed with the decora (a double gang 2 device cover plate) 

Each switch has 2 screws for the conductors change them out so they match the existing red on one side black on the other. If your new switches have both terminals on one side it’s ok put the blacks on the top and reds on the bottom you may need a longer jumper for the black but that’s all,  

Newer switches will have a green ground screw on the yoke or metal strap, you don’t have a ground wire so that won’t be used, the metal screws to the box will ground the switch if the box is grounded through conduit or if no ground it will be the same as most homes pre ~1960. After installing the new switches turn the breaker on and see that you did it.

I would use wire nuts on the splices that’s are there but if you find them wrapped and soldered tape may be easier.

On metal boxes I like to cover the terminals with a wrap or 2 of tape around the body of the switch, this keeps the heads from touching if something loosens up. Tape is not required and not all electricians do this but on metal boxes with minimal clearance I think it is a good idea.