How long on average would it take to cycle a 5.5 gallon tank? I use tap water from a drilled well, and we have a water softener system. The pet store told me to get some sort of conditioner for tap water. Does it really make a difference?
Definitely get your water tested. For both your purposes and your aquarium.
When I first started fishkeeping 20 years ago, we had well water also. You will definitely want to get this tested now and probably in a month or so (if it's not too expensive) and get as extensive of a report as possible, at least the first time. That way you know what you're dealing with. The reason to test down the road is to see if your well water quality varies throughout the year. This can make a difference for certain fish.
The problem I ran into was that I got massive algae blooms called 'green water' (I didn't know that's what it was called until now). My well water had certain nutrients in it that would cause the water to go murky so bad that I couldn't see from one end to the other (4 feet in a 55) within a week or two after I added any water from the well. I had to run a diatom filter w/diatomacious earth powder to filter out the green water, and then figured out that running it just after a water change took out whatever caused the problem.
Our well water had a lot of iron in it, and the well head was near a sulfur deposit, so every once in a while the hot water smelled like rotten eggs. Eww.
If you can, I suggest going to a LFS (local fish store) and asking them to squeeze out all the gunk out of one of their sponge filters. Usually you have to FIND a store that has a few of these, most of the box stores like Petco use centralized canister filter systems. Get the grocest, nastiest looking bag of water you can, dump it in the tank, and you will have an immediate bacteria colony. It will still take a couple weeks for them to fully adhere to all surfaces, etc, but you will not see any chemical spikes because bacteria exists to eat up the ammonia and nitrites. Just don't over-clean for 2 months and you should be fine, water changes to keep Nitrates in check would still be in order.
PS well water tends to be high in Nitrates too. Also read up on the relationship between water hardness (KH) and alkalinity (PH). Just google 'aquarium ph kh'. In fact, google anything on aquariums, it's easy and smashing good fun!