The new Katy trail provides excellent access to downtown for commuters. You can take a bike on the light rail trains at any time of day (not just off-peak hours). you can take a bike on almost any bus, space permitting. You can even take a bike on the Trinity Express train to get between Dallas and Ft. Worth. I believe DART has plans to install bike racks on buses, but I have not seen them yet.
The bike route map is actually quite extensive for most cities. Unfortunately, it hasn't been updated in many years so there are some spots that don't work anymore, but that's pretty minor. I have found the routes to be reasonably signed, though it doesn't hurt to take the book and familiarize yourself with routes new to you.
I have ridden extensively in Dallas - not necessarily on bike routes, Plano, Richardson, and cities east, and I have little problem with motorists except in transition regions between (sub)urban and rural areas.
Dallas has plans to further extend their bike path network, though these things always take more time than seems necessary.
I'd say that for a city that is so automobile oriented, Dallas is actually doing a pretty good job.