When I epilate or use a razor I usually apply alcohol gel afterwords, sometimes helps a bit, but I doubt there's little you can do to entirely prevent bumps when hairs start to regrow. Exfoliation might help.
Cleanse well first, and use very hot water while shaving. Cold water afterwords to close the pores.
And trust me, none of those "miracle products" work better than alcohol gel which is very cheap from our local discount store.
xx
Laser, epilation or waxing might be alternatives for you, although some of these will incur additional costs.