I'll explain multi-civ cards first since that's relatively easier. They're summoned/cast just like normal cards, except that you need to tap at least 1 mana of each civ. For example, when you summon Skysword(light+nature, cost 5), you tap at least 1 light and 1 nature mana and 3 any other mana.
All multicolored cards are put into the mana zone tapped, ie. you can't use that mana on the turn you charged it(it untaps next turn like all other mana).
Cross gears, they are summoned like creatures(called generate instead of summon). You pay the mana cost and generate it. It just remains in the battle zone. Once in the zone, you can pay the cost again to attach, ie. cross it to a creature and the crossed creature gets the abilities written on the gear.
For example, if you want to use Shining Defense(Cost 1 light gear), tap 1 light mana to generate it and put it into the battle zone. Then, when you have a creature in the battle zone /, you can tap 1 light mana again to cross it to that creature.