Currently, you can battle Barbarians and other players for tiles and your cities can be raided by Barbarians. There is no option to attack another player's settlement. After a battle, you get a "battle report" in your activity stream that shows the initial numbers and losses of both armies.
To engage in battles outside of your city, you need military units and the Parade Ground building. The higher the level of the Parade Ground, the more armies you can send out.
It is wise to make your army from different types of units, as they all have their strengths and weaknesses. For example, an army of only archers or artillery is most likely going to be wiped out before they can kill a single enemy unit. Likewise, if you only include infantry, they will receive heavy losses from enemy arrows and shells before they get to do anything.
However, another thing to consider is marching speed. This will be decided based on your slowest unit. Therefore, including artillery in a campaign to a faraway tile will severely slow you down. Marching speed can be increased with the The Steam Engine wonder.
There is a difference between "raid" and "capture". Raiding means attacking the tile and returning home (if alive) while "capture" means your units stay garrisoned on the tile, earning you a constant boost to resources or culture. To successfully capture a tile, you need at least one non-air non-artillery unit surviving.