Since you don't plan on a filter it will be difficult if not impossible to establish the nitrogen cycle due to the limited dissolved oxygen content that the filter will help provide as the water hits the surface creating the gas exchange....Anyway...
In your 4-5gal unfiltered tank-using your thermometer- equalize the water temp at the tap to 76-80F and fill the tank-Add proper amount of dechlorinator-give a slight stir-then properly acclimate the Betta and snail and add to the tank...
To properly acclimate to both the new chemistry and temp-while the Betta is in a small holding container-dump half that water out and add small amount of the tank water until full-Repeat-Then either net the Betta or pour the Betta into a net and add to the tank.
You don't want to add any of the pet shop water to the new tank.
Have some extra like temp dechlorinated water premixed to use for a top off if needed.
Acclimate the snail in the same way....
What kind of live plants, what kind of substrate and most important...Lighting-kelvin and watts of the bulb-do you plan on any natural sun light.
You may or may not need a heater, however, its a good idea to have on on hand-your goal is to maintain a somewhat stable temp in the 76-80F range.
You will need a thermometer to monitor both the tank water temp and the temp of the replacement water used for water changes.
Depending on the number, specie and growth state of the live plants-often they will provide the needed oxygen to help establish the nitrogen cycle and you may not want to make any 100% water changes like you normally would need to make on unfiltered tank.
On about day 3 of adding the Betta, snail and plants-make a 50% water only change...be sure to try and get the water temp of the replacement water within at least 10 degrees of the tank water temp and use the proper amount of dechlorinator.
Once we have more info on the number, species of live plants and lighting-We can give you a better idea of water change needs to maintain water quality....
Be sure and post a pic once you get it set up.....