Leveling up skills is essential in skyrim because as you get more powerful enemies are easier and easier to slay. Especially on harder difficulties leveling up is almost a necessity to have an easy time with the game. To gain those vital increases in magicka, health and stamina you have to level up individual perks as fast as possible, and putting points into some perks makes a huge difference in the long run. Below is a guide that will help you acquire skills faster and thus speed up the leveling process.
The fastest way to level any skill is through training. You will want to train every level because you can only train 5 times per level and later levels take a very long me compared to the earlier levels. You will want to find the appropriate trainer for the particular skill you are looking for. As a general guide you will want to go the the mages college for training in any magica skill, the companions guild for weapon and armor training, the thieves guild for pickpocket, sneak, lockpicking etc. Usually specialized vendors also offer training for their specialization, for instance a smith will usually offer smithing training as well as selling items related to smithing. You can always pickpocket the gold you have spent on training from the trainer, but larger sums of gold usually have a 0% chance of being pickpocketed.
The other advice I can give you is to use different ways to achieve things. For instance if every time you go low on health you use a healing spell your restoration will increase and thus help you level up. Also if you pickpocket often, which is incredible easy to level up, you will not only be able to sell those items but also quickly go through the levels. I also suggest sneaking around in dungeons as it can make some fights much easier if you can start the fight off with triple damage and bring down the health of an opponent by a fair way. It is also fun to murder people without them ever knowing you are there.