Cycling your tank saves you a little work. But, more important, it provides cleaner water for your fish
Thee are two ways to cycle:
Put your fish in the 5gal and do a fish-in cycle which uses the fish as the ammonia source. I won't explain it further because I believe it is not kind to the fish.
Better, in my opinion, is to put your fish in the 5gal and do 2x50% water changes per week to keep the ammonia well below 0.25ppm. You'll need a liquid test kit to keep an eye on things and to monitor the cycle.
First of all, the most important place for the Beneficial Bacteria (thenitrifying bacteria that eats the ammonia) is in the filter. When you cycle a tank, you're really cycling the filter. Gravel, tank walls, decor, plants all help, but the filter does most of the work.
If you have access to filter foam or gravel from a cycled tank, that's a plus. A quarter cup will do. Put it in your filter. The BB in the media will kickstart your cycle. It's OK if you don't have it; it just takes longer --- a lot longer.
You'll need ammonia to feed the BB. Some keepers use raw shrimp or fish food which rots to provide the ammonia. That's smelly, messy and can grow dangerous mold. Find “pure” ammonia that does NOT foam when you shake it.
OK Here we go....
Setup your filter and heater in your 1gal bowl or even a bucket.
BB like air, food and warmth (don't we all?) so:
---Run an airstone if you can. Or use a sponge filter to accept the cycle
---Run your filter full-blast
---keep it warm, up to 90* is OK
---keep it dark; the BB like that
--- add enough ammonia to get a reading of 3.0/4.0ppm on your test kit.
Maintain this reading. Monitor your parameters. When the ammonia goes down, the nitrites should go up. It may take weeks or longer. Keep the ammonia at 3.0/4.0ppm. When the nitrItes go down, the nitrAtes should go up. When the ammonia = 0.0ppm and the nitrites = 0.0ppm, you should see a high reading for nitrates (~40ppm is not unusual).
Then dump out the water and refill it. Don’t let the filter get dry. Bring the ammonia up to 2.0/3.0ppm again. If ammonia, and nitrites both drop to 0.0ppm in less that 24 hours, your filter is cycled.
Place the filter in your show tank with your fish, and monitor it carefully for the first few days. Basically you're running a cycled tank from then on.
I just read this over. It doesn't sound too complicated to me. But, if this is the first time you're seeing all this information in one piece, it might be a little intimidating. Read it slowly again; read other things about cycling. Let me know if this makes sense to you. Ask questions.