Here are some great ways to train your combat. I've made a few adjustments to the article and added an extra method of training combat.
Levels 3-20
Alright, at level 3 combat you have two options.
First, you can kill chickens, which are located near the cows in Lumbridge. To reach there, cross the bridge, keep following the road and just next to the cows there's a small place where you can find chickens to kill. They drop feathers, raw chicken and bones.
Second, you can kill cows which, in my opinion are best for training at low levels. Cows are found across the bridge outside the Lumbridge castle, and outside Falador south entrance, near the Yew trees. They drop bones and cowhide. Cowhides can be good money for new players. They can be sold for 100 each or more in the Grand Exchange or to other players.
Levels 20-35
At this level, fight Al Kharid Warriors in Al Kharid, or you can keep killing cows for money as well as combat levels.
Levels 35-50
Well now, you have gotten stronger and can fight more powerful monsters. For these levels you can train with Hill Giants in Edgeville Dungeon.
You can reach there from two places, from the trapdoor in the building near Edgeville bank (near the Yew trees), and from the small house west of Cooks' Guild.
Hill Giants are level 28. They drop pretty neat stuff like big bones, laws, natures, cosmics etc. The only drawback is that the Hill Giants' area is usually crowded with people.
Levels 50-76
Now you are strong enough to kill Elvarg the Dragon (level 83) and complete Dragon Slayer. After you are done with the Dragon Slayer quest, I suggest that you buy full rune.
Now you can train in Crandor or Karamja Dungeon. Don't get too excited and train with Lesser Demons because they are very strong for you now. Train with Moss Giants instead, as they are level 42 and easy enough to train with as they have a low defense. They give big bones, and cool stuff like laws, natures, deaths, cosmics, gems etc.
You may think that training with Lesser Demons is faster but you will need the prayer exp from Moss Giants so that your prayer level doesn't fall behind.
Levels 76 and onwards
Congratulations if you have reached this level. You can now train with Lesser Demons which have a combat level of 82, or you can train with Ice Giants when you're level 90+.
Now, people may tell you that Lesser Demons have a low defense but trust me, they DO NOT have low defense. Bring along plenty of food, as you will need them. Lesser Demons drop runes, and occasionally rune meds which can be sold for about 10k or high alched for about 11k.
Another way to train when you are level 90+ is to kill Ice Giants in Ice Cave, south of Port Sarim. I personally think it is better to train with giants because of the big bones they drop. Big Bones are essential if you want to train prayer as an F2P player. It is recommended that you take full rune and a rune weapon. Be wary of any Ice Warriors nearby since Ice Giants are often seen together with Ice Warriors, which tend to gang up on you. Ice Giants drop items pretty similar to what the Moss Giants drop, with some additional items such as Black Kiteshield.
I hope this guide has been of help to you. Feel free to comment and let me know what you think about it.