Yes, you can transfer money from a taxable non-retirement account in your names into a UTMA/UGMA account for your kids. Any amounts transferred in a single year over the annual gift exclusion amount would require filing a gift tax return and would eat into your lifetime unified exemption amount. If you ended up filing a gift tax return, you or your executor in theory should keep track of that for after you die and accounting for that properly, which I think is a pain. Since it sounds like you're married, that would be $30K per year per kid ($15K from you, $15K from your spouse to each kid per year), so probably not an issue if you're careful about the transferring.
Money transferred into a UTMA (almost all states are UTMA now, BTW, IIRC; UGMA was an older law) becomes the kids' money and spendable by them on whatever at age 21 (or younger in some states). You can't legally, therefore, take money out of a kid's UTMA account and spend it on yourselves (like pay your electric bill or buy yourself a car).
Whomever's name is on the account is where the income will be taxed. You should be aware of the "kiddie tax", which prevents high income parents from storing money in their kids' names to get lower tax rates on the income. What happens is any unearned income of the kid is taxed at the parent's rate after I believe the first $2,200.
As Tyler Durden says correctly above, money in the kids' names is more heavily "FAFSA taxed" if they go to a FAFSA-based university (most are FAFSA except some select liberal arts and select "famous" schools).