While unlocking a door, press UP on a key, then quickly pause game play afterward. Look at the tab you raised up. If it is mostly up and about to reach the top, resume the game and do it again. If it is mostly half-way or mostly down, resume the game and quickly press X or B. You should have that tab done. Continue on with the next tab. This may require a couple attempts, but after you have done it once it should be easy.
Open locks more effectively by listening closely to the sounds of the clicks the tumblers make when you hit them. Try hitting them when they click twice.
To easily break into a house or chest, pay attention to the speed of the tumblers when they drop. If you try after the tumbler fell fast, you will have a higher chance of keeping the tumbler up.
The Skeleton Key is an unbreakable lockpick that also fortifies your "Lockpick" skill. To obtain it, perform a simple task for Nocturnal, a Daedric Lord. To activate this quest, go to the Leyawyn/Bravil area and open your map. There should be a river running vertically down your map. Look just east of the river and find the black line on your map. Follow this road north and check your compass on either side of the road for the "shrine" marker. You must be at least level 10 to do this quest.
Get into the university, then obtain a basic "Open Lock" spell and train your alteration. Then, make the spell of the maximum possible strength.