For ethernet, I would go with Cat5e, unless you are running some sort of mega-LAN with multiple video clients inside your house and need fully gigabit ethernet. Cat5e will support gigabit at shorter lengths, as long as it is not picking up too much digital, video, or power line noise (keep them separated). It's easier to work with and cheaper than Cat6. Plus it is more than adequate for internet streaming video or audio.
For video or audio feeds (signal level), RG59 or RG6 is probably all you need. I use a form of RG59 (Canare LV77) for all my interconnects and have RG6 in the wall for cable or other distribution. Forget the screw on tips and get a good compression crimper. For 59/6 they are not expensive and you can pick one up at the hardware store.
Speaker is way too ambiguous. We'd need to know distance, power, how many channels, in-wall. For the most part, you can't go wrong with 14AWG CL2/3 rated in-wall, but again, if the runs are long, or you have a lot of power going to them, then you may need to step up a gauge (or two).