How critical is price, as I've sourced 4 layer PCBs from China, where the price is minimal. The problem with modern digital devices is not so much the ultimate clock frequency, but device rise times. These coupled with EMC compatability requirements make a multilayer board a requirement for any digital design, with at least one contigous ground plane. This is basicly my opinion, but years of trying to get products through EMC etc has biased my view, so much that it is 2002 since I last did a 2 layer digital board.
The most important part of routing a PCB is COMPONENT PLACEMENT. Place the compoents and look at the paterns formed by the rubber banded connections, try to create as many point to point connections as possible. If you are stuck on a two layer design, use the compoent placement to minimise the use of vias. Placeing discretes can be a big factor in achieving this, so dont restrict the orientation of SMD discretes.
A big part of being a PCB designer is laying out boards, you develop an eye for placement and subsequent routing from looking at the connectivity, this is learned by experience. So the best way is learn the basics rules, clocks first etc and try it out, route the design and look for improvements as you route. It is an iteritive process.
Auto-routers like all tools require using correctly, again you learn to get a feel for them the more you use them, and the more you learn about adding design constraints.
So welcome to the land of PCB design, it is a world of PAIN