I think most important is to know what kind of mix you use and the drainage. Mold and 'other stuff' sounds like your mix could be too wet. I use a quality potting mix that contains perlite and fine pine bark for drainage. I definitely won't be watering everyday for indoor growing and would let the top of the mix dry out a bit first.

I use the 16oz red cups which is pretty popular and poke 2 or 3 holes for drainage. I give the plants a light diluted liquid fertilizer once a week.

I agree with max. I use anything that is available and usually end up in 18 oz regular red Solo cups. The action is in the soil, light and drainage more than what it is wrapped in. But I've used everything from seedling flats, dozen egg bottoms, jiffy pucks, cut milk cartons and containers, perlite, diatomaceous earth, rockwool, and at the end of it all, nothing beats the economy of 1020 flats (around 50 plugged) transferred into Solo cups which I then top off once as they grow. It is really hard to have bad luck once you get the soil, moisture and light formula pat. Here is a Sungold I sowed 31 days ago, using expired Dec 2013 seed. Two of three germinated and one of those two was blind but then sent two side shoots up from each seed leaf and is really trucking. At 31 days it is 5-6", oh, and budding for the last week, because they have been subjected to 100 F soil temps stress at times but not long enough to kill them. Here is a pic the only normal seedling of the three, 4" tall, where you can note the buds. I expect it will have ripe tomatoes in about DTM 42 days from transplant (tomorrow), or under 75 days from sowing due partially to its suffered upbringing with all the good ingredients :-) The perspective in the pic is off, it really just scratches 4" tall from soil level, but 3/4" has been added burying some stem to the seed leaves (removed) for the last several days. Also note I use saucers (cut gallon water jug bottoms) and perforate the bottoms of the cups

PC

This is my first year starting tomatoes from seed, and red plastic cups have worked great for me too.

when you grow indoors the amount of you water is just as important as what type of grow medium you use. for an established plant, 1 oz of water for a 16-18oz size pot every 3 days is more than enough, seedling even less, it makes all the difference in plant health. add lots of per-lite and make sure there are plenty of holes in the bottom of the pot which helps minimize the effects of over watering. i wouldn't recommend starting seeds in anything bigger than a solo cup, more chance of holding too much moisture. even then, a solo cup is a bit too big for my taste for seed starting.

only thing about them is they can only sustain the plant for about a month before they need to be transplanted.. so if you start your plants 4-5 weeks before going outdoors they are fine.. otherwise i would suggest using maybe starter trays or 8oz cups then transplanting into a 6in. pot after a 2-3 weeks. this will buy you extra time they can stay in the pot if it turns out to be a cool spring where plants have to wait a bit longer before going out.

being a little anxious i started my plants late February due to an unseasonable warm month and ended up transplanting 3 times (8oz tray, 18oz cup, 6in pot). so i'd advise going straight to a 6in pot (if coming from 8-12oz/3inch pot) and transplant twice rather than three times (more work, plant stress, etc)..

Occasional success doesn't mean that any containers are the proper growing container. If you look at the cell packs or trays meant for starting and growing serdlings the holes/slits allow for very rapid drainage. For a solo cup to perform similarly the base would be riddled with holes and they would extend up the sides. Even then, any container for starting seeds should be much more shallow. Two inches of media is plenty for germinating seeds.

Whenever I go near Lowes and Home Depot I check around for empty containers. If you're polite, they'll usually give them to you, but only after they take any dying plants out of them. They won't give you the plants that they are going to throw away, but they are usually pretty good about seeing the containers reused.After a few years, you can have a good number of them saved up.

Also, I find that certain ricotta cheese containers have excellent dimensions, as well as one brand of dip. The dip is a little wider than it should be for its height, but it's not a deal breaker by any means.