If you really want the fastest and simplest way to fill soul gems, use your companion. I started by enchanting Lydia's bow and one-handed weapon with the soul trap enchant. I gave her a batch of empty soul gems and then simply quest as usual. In just one dungeon, she filled up more than 20 soul-gems! Granted I had to stop and recharge the mace twice during that time, and I generally waited for her to be hitting an NPC before I finished them off, but it worked very smoothly. I noticed the bow didn't seem to use up charges --- maybe that doesn't work because she'd have to be too far away from the target. It was far better than wasting that enchant on one of my Daedric weapons or trying to time soul trap.

I've found the best way to do it was with a dagger enchanted with soul trap for 1 second (I'm playing an assassin/spell-sword character). I have the DB gloves that double all backstab damage and have the x15 dagger damage perk. Since I 1-hit almost any normal character in the game, soul gem is filled when I attack almost anyone. If my 1-hit somehow doesn't work, then I'll dual wield an axe and a dagger and just make my final hit with the dagger. It's worked very well for me so far.

When I enchanted my dagger I used a grand soul so I have something ridiculous like 500+ uses for the enchantment on my dagger so I've only recharged it maybe once during my whole game. This tactic works really well for me and I generally have about 30 filled soul gems when I come back from adventuring. I'm level 92 (or so) in enchanting.

Same here. I enchented a glass dagger with Soul Trap. It only weighs 1lb so it doesn't take to much space and I usually just whip it out right before I'm about to kill something. Be careful though killing small creatures like deer, mudcrabs, rabbits, etc to fill soul gems. Say you only have an empty lesser, common or greater soul gem on you. The small creature will go into your next available soul gem and you'll pretty much have wasted a large soul gem on a creature or animal that wouldn't even completely fill a petty soul gem.

Are companions still bugged on this?

I noticed Lydia had no issues filling many more soul gems than the kills added up to in 1.1, not tried it since.

If you fill a large soul gem with a petty soul or lesser simply drop that soul gem from your inventory and pick it back up. The soul is removed from the gem. I do this with black and grand gems that somehow get petty souls in them.

I keep a lightweight, yet fairly powerful, weapon with a 1 second soultrap. Beat them down with the main weapon and switch over the the soul trap one for the kill.