OK, the R16 is a standard def DVR, and it doesn't have component outputs. The best output it has is S-video, but it sounds like your TV doesn't have an S-video input (most newer TVs have started dropping it). So they best you are going to be able to do for now is the composite outputs, yellow for video, and red and white for right and left stereo audio, which I'm guessing is what you are using right now.
Since you are a new customer it is very unlikely that DirecTV is going to give you much of a deal on upgrading to an HD-DVR. So the $199 up front lease price is probably the best you are going to do for a while. After about a year there is a chance that they may give you a discount to $99 or free, but it will come with a new 2-year commitment. You could talk to them about getting a HD receiver as they are only $99, but then you will miss out on being able to record stuff.
This is exactly why I recommend all new DirecTV customers get HD equipment from the start, even if you don't plan on getting an HD TV right away. The deals on equipment you get as a new customer are better than pretty much any deal you are ever going to be offered again. Not to mention the HD channels look way better, even on an old SDTV (you can watch the HD channels and either crop or letterbox them).