Wear long sleeves and long pants, be sure to leave your shoes in the designated area and ritually wash your hands before going inside, and try not to be too loud. Other than that, basically just copy what everyone else is doing. Usually the statues are organized in a particular way, and you just make a full round through the temple, praying before each one.

Prayer itself is mostly just putting one's hands together, but you will also see people prostrate themselves on the ground and touch their head to the floor 3 times. Also, during prayer to Ganesh it is customary to cross one's arms in front of the head and knock the forehead with one's knuckles 3 times for good luck. Don't point your fingers at the statues, it is generally considered disrespectful.

At some point, you might see a tray with red ash, white ash, water, and perhaps some sweets/fruits. The white ash is meant to be taken with the 3 middle fingers and applied accross the forehead (rub the excess on the arms or neck). The red ash is literally the stereotypical red dot thing; take it with the 4th finger and apply to the forehead after the white ash. The priest will hand out the sweets and dispense the water with a spoon, be sure to accept it with the right hand only.

During a puja (ritual service), remain seated and listen to the prayer songs. The priest will eventually pass around a plate with a ritual fire. This is not a tithe collecting plate, it is for worship. Take both hands, bring them close to the fire, then bring them close to your eyes, and do this 3 times.

Edit: expanded!