Hi there!
Where in the southwest are you? I'm in San Diego.
Yes, go ahead and get started on your Roadmaster. As said above, a new-bought schwinn probably won't be significantly better.
One of the common recommendations to get started commuting is to search your local craigslist for 90's era rigid fork mountainbikes, and put city tires on it instead of knobbies. Those bikes are bombproof, plentiful and cheap on craigslist in most areas, they usually have mount points for racks, etc.
Search your local CL, see if you can find any bikes that look interesting to you, and post links here, we love to look at bikes and give opinions!
The thing about CL though is that you have to be prepared to possibly either put in some elbow grease of your own, or pay for a shop to tune up an old bike. If that doesn't sound like something you want to do then maybe you want to buy a new bike from a local bike shop.
Or a middle ground would be a new bike from online, which you would have to minimally assemble. bikesdirect.com is pretty popular. You might take a look at
city bikes, or maybe a cheaper
mountain bike (hardtail, not full-suspension, and again, you'll want to change to smoother tires)