There are plenty of paths that you can follow once you leave the Asylum, but most of those will just get a new player killed. They do open up possibilities for more veteran players though.
As you point out, the game is old, and feels old if you've played any more modern souls-likes. Still, the first one is the only one I've played of the series. I have played lots of others that are not part of the Souls trilogy though.
As for the Taurus Demon, there are a few options. First, do you have any gold pine resin? You need the skeleton key (I think) to get this at this stage, but if you have it, use it, it's extremely powerful against Taurus.
Second, do you have any black firebombs? Also very powerful at this stage.
Third, did you climb the ladder next to the fog gate you exit just before the Taurus Demon? There are 2 xbowmen up there and they should die before you do anything else. Once up top is clear, trigger the Demon then run back and climb the ladder. Plunging attack that dude for massive damage. Plunging plus gold pine will bring him past 50% with a single attack.
Fourth, find a weapon with bleed (morning star, dagger, thief's dagger, scimitar, etc) and just build up bleed on him, he's very vulnerable. You should see large chunks of his health disappear after a few swings. Added bonus, daggers and scimitars are very fast, helping you get in and out with less risk.
You can actually keep your distance pretty easily in this fight, so spam any magic you might have, bombs, knives, whatever. You can wait for it to over commit and then run/roll past it and climb back up the ladder if you are fast, then plunging attack it again.
Or you can plink him to death. Wait for him to over extend, run in and whack him, then run/roll out again. Over and over. Of course this could take awhile and the longer you fight, the more likely it is that you'll make a mistake.
