All of the combat tutors may be found just north of Lumbridge Castle.
Magic and Runecrafting Tutor: Mikasi
Mikasi, like most mages of any significant power, likes to teach people the ways of the runestones. Having spent many years herself learning the finer points of crafting runes and the rituals that bind the Magic to the stones, she has returned to Lumbridge to spread the wealth of her knowledge among the less experienced. More talented practitioners of the arcane arts will find that Mikasi also knows much of use to them.
Free items: Mikasi will give out 30 mind runes and 30 air runes - although these are held on a thirty minute timer, so you will have to wait half an hour before you can collect any more runes from her. You will only get runes of each type if you do not have any of them already. While you cannot collect runes from Mikasi and the Ranged equipment from Nemarti at the same time, you can wait half an hour before going to Nemarti. To collect your runes, right-click on Mikasi and select 'Claim'.
Melee Tutor: Harlan
Harlan is a warrior of great experience with a kindly personality. His particular interest in the wellbeing of young adventurers has led him to Lumbridge, where he graciously helps out young warriors looking to train in the honour of close combat. Though he specialises in helping very young warriors, he is also well versed in techniques and locations for more advanced training.
Free items: Harlan will gladly equip adventurers with a training sword and a training shield. He will not, however, give them to anyone who already has them (there is no sense in handing them out to someone who already has them, after all). The training sword is the equivalent of a bronze longsword, and the shield is the equivalent of a wooden shield. Neither of them is tradeable.
Ranged and Fletching Tutor: Nemarti
Nemarti is a skilled archer and capable of drawing an arrow to its target from a hundred yards while at a full sprint. Unsurprisingly, then, she is almost the ideal tutor for those who want to train in the arts of the rapid strike from afar. Further, she is also skilled in the art of bow and arrow making, and will gladly explain how to go about this, provided you are a member. Like the other tutors, she will also provide advice for more advanced students seeking to overcome the next hurdle in their training.
Free items: Nemarti and Mikasi, as is common between rangers and mages, do not quite see eye-to-eye; they do have an agreement, though, that they will not offer their wares to each other's students. While you cannot collect the bow and arrows from Nemarti and the Magic equipment from Mikasi at the same time, you can wait half an hour before going to Mikasi. Nemarti will gladly supply anyone who visits her with a training bow and 25 training arrows. This bow can only use these arrows, and the arrows, equally, may only be used by this bow. Like Mikasi, she will distribute arrows and the bow only every half hour, and not to anyone who already has a training bow or training arrows. Like the training sword and shield, these are untradeable.