I started riding seriously about September. I rode a 1993 Roadmaster mountain-type bike that I affectionately referred to as the "POS". I upgraded the tires to road slicks since I didn't ride off-road at all. I had the new tires on for about a week before I bought a used road bike. All in all, I rode the Roadmaster POS for about 3-1/2 weeks. I was absolutely SHOCKED by the difference between it and a better quality bike.
Nothing wrong with starting out on the RM, but I'd not suggest any real "upgrades" to the bike itself. If you get hooked anything like I did, the Roadmaster will not be around long enough for any upgrades to be of any value.