There's a substance called tannin, which some people might have heard of in context with wine, present in all woods, including the woodier parts of a leaf like a tea leaf. It's acidic and bitter. (It ends up in wine because the wine is stored in wooden barrels and leaches it from the container.)
What happens if you chop up a leaf into small parts or, worse, dust? You end up with more tannin. So leaf size can be important. There is a standard rating system that gives you an idea of the leaf size, or you can simply buy whole leaf tea that you can see.
Darker teas are more oxidized, usually by roasting. They will usually be more bitter by design. My favorite black tea is Assam, specifically Ahmad's Khalami Assam, which I get regularly for like $12/pound on Amazon. It's probably underpriced; it's mostly full leaves and includes the tips, which have more aromatics. I can drink it black but I prefer it with milk.
Milk, in general, is a good way to reduce the bitterness of tea, but it doesn't work if you've got a floral tasting tea (I'm sure you can imagine that isn't very good) like most green teas. (I do like to make Kashmiri tea, though, which is green tea brewed for a very long time and then diluted with milk.)
If you like the green tea you're drinking and want something even milder on the bitterness/acid front, you can usually get some good quality whole-leaf green or oolong teas at an east Asian market if there's one near you. Sensha is awesome if you can get it fresh (the leaves will be visibly shiny needles), but it's expensive, so obviously not a daily drinker. With pretty much any whole-leaf tea, about $0.50-1.00 per ounce you are getting something quite a bit better than a teabag. You won't necessarily get great tea, but even expensive tea is cheaper than coffee.
The trick with most Chinese teas is that you might be expected to prepare it Gong Fu style, which aside from helping you kick butts helps you make a good cup of tea:
First, the longer you steep the leaves, the more tannin you can leach out.
Second, the hotter the water, the more tannin you can leach out.
Both of these can be avoided by brewing at a lower temperature (about 90C is fine) with more tea leaves, and brewing small batches several times. (Think three or four pots of tea out of a single bunch of leaves.)
There's not a whole lot you can do with a teabag, but I do find that if you use a lower temperature and a short brewing time that you can get a bit more out of them.
If you are a former coffee drinker but you want something that reminds you of coffee, try Pu Ehr. It's called black tea in China, but it's aged. You buy it in cakes usually and break off a corner. You can't see through it and it can be very strong tasting, but it lacks the acidity of coffee and usually is low enough in bitterness that you can drink it black. Just be careful. It's like buying wine, there's some BS behind it and the prices don't necessarily reflect quality. Again, best place to go might be an east Asian supermarket.
I like tea, you might have guessed.