Paul, if all you're doing is DLR and not leaving the resort area, you absolutely do not need a car, and DRE can be a good option. Everything is easily walkable within and adjacent to the resort. There is an Alamo car rental on property in Downtown Disney if you decide you want to take a day and explore L.A., and the same Gray Line operator that runs the Disneyland Resort Express also operates bus tours of the region.
I've stayed at DLR several times with no car, taking DRE back and forth from the airport (in my case, LAX), and it couldn't be more convenient. The DRE buses pick up from a clearly marked area of the common ground transportation curbside (buses and taxis), and drop you off either at your resort-area hotel or at a common stop within easy walking distance of your hotel.
If you're booking trough Disney, you can purchase your DRE transfers as part of a package and they'll be mailed to you along with your park tickets and hotel voucher. (You'll also receive a printed copy of the same DRE brochure with stop locations and pickup times that you can download as a PDF from the links others have provided above.) If not, you can simply pay with cash on the bus. Just download the brochure PDF and plan to bring along enough cash depending on the airport of your arrival (LAX or SNA.)
The only real difference between DRE and WDW's Magical Express is that if you're a group of four or more, a taxi or SuperShuttle-type of service could be cheaper (because, as you noted, DRE is not free.)