It's all in bytes (small b)
Cat5 = 100Mb/s
Cat5e = 1000Mb/s
Both if which are a lot faster than your 20Mb/s internet so for internet use either would do just fine. If you are copying file about between machines a lot then the faster local link the better.
However...
You'd need gigabit rated switches (router) and network cards to get 1000Mb/s from any machine...
However2
300Mb/s is nowhere near what you'd get on wireless in one direction at any time.
Basically.... stick with what you have will be fine for internet, if you want fast connections between several computers on one site then Cat5e (or 6) would be best as long as all NICs and switches are fast enough. (and if you want fast on the LAN get rid of wireless all together, wireless is for mobility, not speed)