Combat, one of the most celebrated of all skills in Runescape, and arguably the most fun. Many quests require you to have a good combat level so it's as important as any skill (Especially if you plan to player kill). So, on with the guide.
I'm not going to talk about weapons too much, just good places to train your combat.
When you first start out you'll probably want to kills chickens in Lumbridge for your first few levels, no further than level 3 though.
After that, you should move onto the goblins behind Lumbridge castle until you're about level 7.
From there, make you're way to the crafting guild (South East of Falador ). Here you'll find the cow pen. Kill these critters until you're around about level 20.
From there, you'll want to go just east of the cow pen onto a little peninsula, which is the home to about 10 hobgoblins. Kill these until you're about level 40.
Once you reach this level, you should go to Falador castle (Just East of the West bank), here you'll find level 43 white knights. Make sure you bring plenty of food with you as you'll be spending some time here.
Once you reach around about level 60 it's time to move onto the big kills, lesser demons. For these guys you'll want to make sure you've got the best armour you can wear and a LOT of food with you. You'll have to travel to Karamja volcano. Do down into the volcano and find the lesser demons. If you can teleport I suggest you bring some runes to teleport out of there, should you suddenly find yourself overwhelmed.