The RM 20 rim was a standard size/model for many mountain bike models years ago, since you have the sidewall numbers off the old tires you can pretty much bet almost any 26"x 1.5" through 2.25" tire will work. DO NOT buy any tire that has a fractional ( 1 1/2", 1 3/4") size designation, only the decimal designated tires will fit your wheels/rims. For your intended(mostly road) use a smoother tread pattern(as opposed to the "knobby" pattern on the existing tires) will be best for you, much smoother riding and less rolling resistance(easier to pedal).
My personal choice would be 26x1.5"with a tread pattern something like this
Kenda K193 Kwest 26 Inch Tire at BikeTiresDirect or this if you really think you will ride much off paved roads
http://www.biketiresdirect.com/produ...s-26-inch-tire but more than likely the former will meet almost all(except for mud, but really, how much mud do plan on riding in?) your needs. Here is another, a 1.75" thats decent:
http://www.westernbikeworks.com/prod...ntry-rock-tire The tires I've suggested(
or similar ones of different brands) are under $20, so you should be able to get 2 tires and tubes for around your $50 max, you can spend a lot more and being a new rider you won't benefit much(probably not at all) from spending more. The tires
bikemig suggested
Amazon.com: Continental Town & Country Urban Bicycle Tire: Sports & Outdoors are excellent tires for $10 or so more for the pair, just depends on how much you can really spend. Remember to factor in shipping charges, some places have free shipping specials or free with minimum order
@Nigel - Reviews seem to say those won't last very long, and are low pressure (I read the reviews for several listings of C727s). Should I be concerned with either of those things?
While I am certainly trying to be cost effective, I am also willing to pay a little more for something that will last a lot longer.
Relative to what? How much do you weigh( light, heavy, mileage will vary), how many miles do you intend to ride a week, 10, 20, 50? If kept properly inflated I'd be surprised if they won't last you at least a year. Low pressure usually means up to 65lbs psi(+-), this should not be a deal breaker unless you are interested in maximum speed with the trade off of comfort. Try not to get hung up on one(or so) negative review, even $80 tires have a bad review from someone, be more concerned with the overall product rating.
I'll make a plug for a local bike store if you can't mount tires yourself and don't have a friend who can. if they are a decent shop, realize you're a new rider, and want your return business they should mount tires and tubes included in the purchase price(ask them before buying), and many shops have entry level tires at around $20 each, tubes $5-7, again call and ask rather than just showing up.
You came to the right place to ask questions, knowledgeable and friendly people here, and generally you are most likely to get better/correct/accurate info here rather than a retail outlet that is geared to selling product, not always with your best interest in mind.
Hasn't come up yet but what brand/model bike do you have?
good luck, Brian