How can I join two pieces of wood at 90 degrees?
How do I connect [1x4 wood] together at 90 degrees while maintaining a solid connection?


If you can obtain a hand-saw (e.g. tenon-saw/back-saw), some sandpaper and some wood glue, you can make lap-joints. I find they are the easiest way for me to make rigid joints in wood. 



To join wood the other way, for a strong joint I'd try a simple finger joint (box joint)




  at best I can get a hand drill, a jigsaw and some vices


I find jigsaws are a good way to make wobbly uneven cuts in sheets of wood. Maybe that's just me. Chewing through wood with your bare teeth is not much worse. A back-saw and a miter-box might be more useful to you if you can obtain them.