Asking about tea is like asking about beer. To people who don't really drink it, it seems like all the same thing, but it is definitely not.
For a "proper cup", that is, strong black tea with no sugar and only a bit of whole milk (not cream), buy Bewleys in bags directly from bewleys.com .
PG tips makes an acceptable proper cup and can be found at better supermarkets.
For black with sweetener, you're basically ruining the tea, so use any English breakfast or Irish breakfast tea, and they will also do in a pinch with milk, but FFS do not put English and Irish teas next to each other in the cupboard. Burn everything British but their coal.
ETA I don't normally do black with nothing, but it is very good to do with Earl Grey or Darjeeling. Look for Twinings in supermarkets. There is better than Twinings but also much worse... They're reliable.
I haven't yet gotten into oolongs.
Going into the greens, first off, everyone should, because they are awesome. Gunpowder tea is available cheap and in bulk at any asian market and they're all roughly the same. But the secret to greens, and probably the reason people don't like them, is you need to get the tea out of the water a MAXIMUM of 90 seconds after it goes in.
The best green tea that I serve for company is Yun Wu.
ETA a lot of what I mentioned here is more expensive than cheap ass Lipton, bit it's still all cheaper than soda. For $10 a month one can drink really enjoyable teas.