The approach you describe is very sound IMHO, it is the most economical way to get multiple roles out of the same gun. Several manufacturers offer models whose barrels are interchangeable for that very purpose. And it is not necessary to buy the set as one unit- look for a good used gun of the make and model of your choosing, and keep a lookout for the other barrel or barrels of the length and style you want- interchangeable choke tubes, vent rib, rifle sights etc. Barrels for modern repeaters are not 'fitted' but will interchange among the same make model and gauge.
First thing to do is to decide what gun you want, by renting different guns at the gun club where you shoot or by shooting different guns belonging to friends. After you make up your mind which gun you want, then start shopping. I find better deals on used guns in good condition, and will always buy an extra barrel when I find one at the right price.
I prefer Remington 870 pumps, they have a 55- year track record for durability and performance. Not to mention that they have been made by the multiple millions. They are also among the most modular and versatile guns available, and easy to set up for whatever task is at hand, the virtual Swiss Army knife of shotguns. They are sold as 'Combo' sets new, as Kudu mentioned, or you can do-it-yourself by shopping for used gun and barrel(s) and save money as well as having more choices as to the barrels you get.
It shouldn't be too hard to get a good used 870 Wingmaster and an extra barrel for less than $300 with some juducious shopping. Sometimes LE agency trade-in guns are available also, often these have been carried a lot buy not shot much. It certainly is doable with Remington's economy Express guns, be sure to look for an older model with an aluminum trigger plate and no dimples in the magazine tube though.
Do stick to a smoothbore barrel for your short home defense barrel, sometimes short barrels are offered in a rifled version for big game hunters who want to shoot sabot-type slugs. You will find smoothbore versions of short 870 barrels offered from the factory in lengths of 18 or 20" with rifle sights or simple beads, some even have interchangeable choke tubes. Remington makes a 21" vent rib barrel with interchangeable choke tubes for its turkey guns, if you wanted to 'do it all' with one gun and one barrel or a while, this would be a great one to get started on.
You are not limited to pumps, by the way- you can do the same thing with semiautos. It may well take more shopping (not to mention more money) to put together a package on the used market, though. And getting an out-of-mainstream brand will make it difficult to find extra barrels and accessories at good prices.
HTH, remember you can ask more questions if needed... and there is a lot of material here about this already if you search for it.
lpl/nc