Okay, first press "P" which is your skill pane. Press the boot icon (general tab). In there you should see your newly acquired abilities, leatherworking and skinning. Press on the leatherworking button and it should give you a few basic craftable items. That's what you should make in order to train your leatherworking up. Also, in order to train skinning, you should skin higher level beasts. The first few levels you'll be able to skin up to like level 6 or level 7 beasts. Later on, the amount of skinning skill you need is 5*mob level.
In order to skin some mobs, you will need to get +skinning gear, which can be acquired through enchants and items. Also, once you hit a certain level leatherworking, you will have to decide into which specialty you will go into. Either elemental, dragonscale, or tribal. Dragonscale goes into mail, so take it if you are a class that can wear mail at level 40. If you can wear plate, i suggest taking up blacksmithing. Elemental is more leather melee gear and tribal is more of leather casting gear.
Also, when you hover over a skill or beast, you will get a different colored text saying skinnable or in some cases, different colors (untrainable skills are red).
Red=Too low leveled; need more skill in order to perform the task
Orange=Good chance of leveling; may fail (sometimes skinning requires skinning several orange mobs before gaining a skill level, because of the abundance of beasts)
Yellow=okay chance to level, it's usually about 50/50
Green=Low chance of leveling
Gray=No skill ups are given