It seems odd to have a section explaining how to level up, but the Elder Scrolls games have never been the most clear in this regard. The leveling system was always fairly obtuse and required a good bit of thought to avoid screwing yourself over near the middle or end of the game. To help draw in newer players Skyrim has fiddled with the finer points of this system, although it still works in a manner similar to older games and there are still plenty of finer points to learn.
Raising your skills in Skyrim requires that you either use the skill often or pay someone to train it up for you. Paying to raise your skills is a nice way to get a quick boost, but it’s both expensive and kind of a waste. Generally speaking, paying for skill raises should only be done if there’s a perk you’re aiming for or if you wish to gain some quick points in a skill that hasn’t been given a lot of attention previously. Improving your armor skill like this is a fairly good idea if you decide to swap what armor types or magic you’re using after the game has begun.
But most of your skill raising will come from simply using your skills repeatedly. As such it is important to repeatedly use the skills you want to train up. If you spend too much time using the ones you aren’t planning on sticking with then you will level up rapidly while your primary skills aren’t high enough to take the really good perks. In this game the new perk system is what really determines how good you are at something, not your (now absent) stats or skill levels.
The Guardian Stones are something that you will want to activate very quickly. There are three of these stones standing around in a semi-circle on the road to Riverwood from Helgen (check your map for the specific location, it’s marked by default). Each of the three stones is linked to the skills of a particular skill batch; warrior, mage and thief. When you have a stone activated all skills from that branch level up considerably faster as they’re used.
In the older Elder Scrolls games, a boon like this would actually turn out to be something of a burden due to the obtuse leveling up system. In this game though, they’re definitely something you will want to use. You may want to consider not actually assigning it to the skills that you spend most of your time using, instead assigning it to one you use a bit less. If you’re playing a fighter who uses magic occasionally then activating the mage stone will help your magic improve fast enough that it will stay effective throughout the game.