It is advisable to pick up a pair of army issue assault boots, perhaps from a surplus store. Soldiers who use them daily criticise them for being uncomfortable, but you will probably be using them once a month so they will be entirely tolerable. Once broken in they will give you great traction and ankle protection and will last for years if cleaned and stored correctly. After market (non military issue) boots in the same price range as assault boots will barely last a year before they fall apart.
Boots are worth the investment to keep your feet dry, warm, increase you grip and protect your ankles. Trainers are a poor choice, especially in woodland games; they ruin your camouflage, look stupid and significantly increase the chance that you will sprain or break you ankle (painfully sprained ankles are the most common injury in airsoft). It is the same as buying a mask; if you buy a gun before you buy boots then you have made a big mistake.