In my experience snails will only breed massively out of control if you are over-feeding the fish, or have a lot of algae in the tank or don't clean the substrate enough or simply have a lot of bottom dwelling/feeding fish in the tank which need to be fed sinking foods etc. Given that, i've always had a pretty large trumpet snail population, but i think that is due to all the cucumber i feed my pleco's which needs to be sunken to the bottom of tank.
With getting rid of snails, you can either pick the snails and eggs out of the tank by hand, or used a copper based fish friendly med to eradicate the snails (however copper based meds will harm any invertebrates you have like shrimp etc).