I first trained myself by repeating 1-1 multiple times. The weak enemies drop enough healing items that you'll basically be set for the entire remainder of the game if you just train on 1-1 for a while. I used a magic build, and trained for a long while, calculating the damage on each hit from my beams, before taking on the Red-Eye Knight who's off the main path on 1-1, but if you're very careful about staying invisible and staying out of his sight, he can be a pushover.
My next stage was 3-1, and even before finishing it, I was taking practice runs through the parts that I'd already mastered so that I could stock up on magic-restoring agents. The Mindflayers drop them in spades.
The very first part of 4-1 will get used to the gymnastic skeletons that can be murder on a beginner. My magic build was able to take them down from afar, and the rest of 4-1 is surprisingly easy.
4-2 is an excellent place to farm souls and also get lots of practice in for the intermediate player. Do the "drop down and hit the Reaper with ranged attacks" thing first, and then you can either take a step backward out onto the ledge (behind you if you're facing the Reaper) and train yourself to fight the skeletons that will come at you along that thin ridge. Fight them and grab another ~2000 souls; worst comes to worst, they kill you and your bloodstain is right there for you to pick up.
It is very gratifying to see your skill rise, and to see those nasty skeletons go from the scourge of your existence to just another enemy that you can handle.
There's a whole lot more to 4-2 after that part, and there's a good chance that you'll die a few more times even after testing your mettle with the skeletons, but you should have picked up enough soul levels to make the nest part a little easier.
Then in 4-3, once you've beaten the boss, you get to have a whole lot of fun with a special weapon that you can only use there. Trust me, grinding on that stage is more fun and stress relief than actual ability honing.
The steady progress that you'll see in your own ability is the most satisfying thing about Demon's Souls. The first time, you're cursing the monsters and the game's developers and God and fate, but the next time you'll actually manage to kill some of those nasties, and before you know it you're walking through those halls and caves with some confidence.
I must have died a hundred times in 1-1; maybe more. But I didn't die a single time in either 5-1 or 5-2, until the boss fight. If you'd shown me a video of my own play in the Valley of Defilement two weeks before I actually did it, I wouldn't believe it. But trust me, you'll improve in just the same way.
Oh, and the final battle in 1-4, which I saved for the very end and only took on at around SL 86, was one of the most thrilling boss fights I've ever had in any video game, ever.
Do not give up on Demon's Souls. It's a great experience.