When I was leveling my Shadowpriest, from level 5-20 I had to rely heavily on a good wand. When I hit 20 and gained Mindflay, I didn't have to rely on it so much, but it was still as staple.
I cannot stress this enough, that a good wand before level 40 can and sometimes will out-DPS your own spells, especially if you have the talent Wand Specialization.
Also, as MookusOU said, Vampiric Embrace, which returns shadow damage in the form of health, will make your life a lot easier. Also, as Mookus said, all classes are challenging until you get about 20-25 points in the tree.
So here is a good talent tree progression that works when leveling, especially when soloing.
Level 9-14: 5/5 Wand Specialization
Level 15-19: 5/5 Spirit Tap (Also when leveling, get as much spirit as you can, that way, Spirit Tap will return a TON of mana)
Level 20-21: 2/2 Improved Shadow Word: Pain (GET IT, gives your SW:D 2 more ticks for free)
Level 22-24: 3/5 Shadow Focus
Level 25: 1/1 Mindflay (GET IT!)
Level 26-29: 4/5 Improved Mind Blast
Level 30-32: 3/5 Shadow Weaving
Level 33-34: 2/2 Shadow Reach
Level 35: Vampiric Embrace
Level 36-37: Improved Vampiric Embrace
Level 38-39: 5/5 Shadow Weaving
At level 40 pay the one gold and respec to: http://www.wowhead.com/?talent=bZZxMhpMxR0t
By this point you should be able to figure out for yourself where your talent points should go.
And as the other repliers said, just keep on chuggin' along. Although it's tough at the beginning, Shadowpriests are extremely fun and effective levelers later in the game.
To conclude, read the Shadowpriest FAQ, and Khalane's Shadowpriest PvE guide a few posts down.
Also, on the O-Boards, there is a post about how to level a priest which is surprisingly quite helpful for the O-Boards.