Buying a house with pellet stove in basement, well water and septic tank any tips/advice?

Ask the owners why the pump was replaced last year, and pull out the filter to see how dirty it is. If it were my house, I would get the pellet stove company out before I purchase it, that way if there's a problem I know I can negotiate that... But I don't know your timing

The pellet stove might hear just the basement or it might provide some heat for upstairs, really depends on capacity and airflow, plus obviously fuel price.

My parents have had a septic system for over 30 years and have literally never done anything to it. I know this is the exception to the rule, but a well designed/installed system can last a long time without issues.

True, although I'd recommend at least getting it checked after several years as lots of factors involved. I also forgot to leave some advice to OP regarding being super anal about what goes down the drain. Throw food scraps, fats, etc into the trash not the disposal.

Pellet stove could be for keeping pipes from freezing during extreme cold, but heating the basement will also help keep floors above warmer & reduce heating bill overall. Just check that flu pipe is safe for future burning. It can vary widely depending on septic size and usage, but getting septic tank pumped out every 5-7 years is a good practice. A new well pump can last decades with no specific maintenance. If any pressure issues emerge you'll want to learn to check your pressure tank pressure. Well systems often have a whole house sediment filter, if so be sure to change that out periodically. I try to do mine yearly but every situation is different.

1st stove with no experience, I would recommend calling a local chimney sweep to have it and the flue serviced and cleaned. I also have a well and septic. Expect a well pump to last 8 years, and have your septic pumped every 2-4 years).

Don't flush anything that isn't TP or digested food (no tampons or an Insinkerator). Depending on household size, pump the tank every two years (or more). Leach fields fail mostly when solids make it out of the tank. Also find out what type of field it is (20 years old - should be a plan on file with the town), the fabric chamber types are extremely vulnerable to compaction (driving an auto or tractor on them). The stove might heat your home, especially if you have some passive vents. A friend of mine heats their 3 BR ranch with a basement woodstove and no active ventilation in climate zone 4/5. Basic artesian well system is pretty simple: submerged pump, and a pressure tank in the basement. If you have a lot of sediment, mineral/hardness, or PH problems, there are systems for those. You'll need to go a season or two before you know it's regenerating capacity vs your usage patterns. LP furnaces are fairly low maintenance (compared to oil fired). Ideally, get it cleaned/inspected annually - but you can probably stretch that to 3+, especially if the pellet stove is carrying the bulk of your heating needs. If it's a high efficiency model, it will have a condensate pump. Monitor that for blockage, and flush it out every year or two (the same goes for your water heater, if it's LP).

Call a well company and have them check out your well system, there should be a pump in the bottom of your well some sort of retention tank in your basement, and hopefully a filter or two. Call a pellet stove company and have them check out your pellet stove. If that is the only heat source in the house, then yes it's practical that that's going to be heating up the whole house, but sometimes these are uses supplemental heat. Having a pellet auger attached to your stove makes life a lot easier if you want constant heat coming out of it for more than 4 or 5 hours.