Rules
Training Grounds won't affect a cost that isn't the cost to activate a creature's activated ability. For example, it won't affect Flameblast Dragon's {X}{R} cost, since that's a cost paid when a triggered ability resolves, and it won't affect a kicker cost, since that's an additional cost to cast a spell. Activated ability costs appear before a colon (:) in a card's rules text, or, in the case of some keywords, before a colon in reminder text.
If an activated ability of a creature you control costs no generic mana to activate (for example, if it costs {R}{R}, it costs {0}, or it costs only nonmana actions such as {T} or "Sacrifice a creature"), Training Grounds simply won't affect it. In particular, it won't increase the cost to include a mana payment of {1}.
Training Grounds affects only creatures you control on the battlefield. The costs of activated abilities that work in other zones (such as cycling or unearth) won't be reduced.
Training Grounds can reduce the amount you pay for a creature's activated ability cost that includes {X}. For example, Drana, Kalastria Bloodchief has an activated ability that costs {X}{B}{B}. If you control Training Grounds and you activate the ability with X equal to 5, you'll have to pay only {3}{B}{B}. This is true even if the ability states that {X} must be paid with a certain color of mana, as Crimson Hellkite's ability does.
Training Grounds can reduce the part of an activation cost represented by generic mana symbols down to nothing, as long as it still costs at least one mana. For example, if an activation cost is {2}{G}, you'd have to pay only {G}. If an activation cost is {2}, though, you'd still have to pay {1}.
Training Grounds takes the total cost to activate a creature's activated ability into account, not just the cost printed on it. For example, Urabrask has an activated ability that costs {R}, and Suppression Field says "Activated abilities cost {2} more to activate unless they're mana abilities." Since activating Urabrask's activated ability would now cost {2}{R}, Training Grounds reduces that cost back to {R}.
Training Grounds won't affect the part of an activation cost represented by colored mana symbols or snow mana symbols. It also won't affect nonmana parts of an activation cost, if there are any.
less to activate. This effect can't reduce the mana in that cost to less than one mana. 
less to cast. 
