via is a little hole, that would be filled with a special metal epoxy at a fabrication house. for DIY, vias are different, they're a much larger hole, that you have to carefully line up, and then solder a piece of wire through
eagle is not the easiest to learn, but it is great software. your design isn't that complex, I recommend you rip-up all the traces the auto router laid for you and do them by hand. try to get as many of them using the blue layer 16 (bottom layer). only use the top layer where nessecary. set your width to 0.02 or 0.024 (24 mil is a standard width in eagle). toner transfer can do 6 to 8 mil but it gets kinda iffy - bigger is better when you're starting out. if you do need to lay a via, set the diameter to 0.06 or 0.07 and your drill to 0.035.
soldering anything except big 'ol ICs and some other random components on the top layer is pretty difficult. it's better to assume you won't be able to solder it, so that means you can't connect to it with the top (red) layer. this will make sense to you if you lay your parts out on a simple print out of your design - many parts cover their pins... with a fabricated board, the special through-plating carries your solder to the top, with a diy board, that won't happen.