Now I know why I suffer so much in early games...In my history with civ5 I've never used tribute option on CS, I'm serious, never. I dont know why but, I've just focused on civ4-kind-of-beggining-style (rush few buildings to have some hammers and gold and avoind military units cos of cost). Well, I'm gonna change it now then.
I'm with you, it took me a while to change my mindset on Authority/Tribute play. I am a builder type player by default, so warring is not my thing. But I will say its really fun to beat up on CS! Its not just tribute, you can milk so much out of those CS.
1) Kill their units, get culture, science, and XP
2) Pillage their lands for gold.
3) Make peace, and then tribute!
4) Rinse and Repeat
5) When you have milked all you can, conquer them!
The trick I found is that you have to really invest in your military early. And for a builder like me you feel so behind for a while, your score starts to sink lower as other civs build up their infrastructure. But suddenly you hit a critical mass. Suddenly your army can pull a tribute from every CS, and you can just tribute tribute tribute in quick session. The bonuses start to flow, and you rapidly catch up and even pass the more passive players. Its a lot more swingy style of play, but its quite enjoyable once you get the hang of it.
I'll also echo CrazyG on Imperium. My default build right now is 3 base cities. I build the settler for the 4th, but don't settle immediately. I then get imperium for my 5th city, and settle both in rapid succession. It bags you a big boost in culture and science. Further, because it instant pops your first border on that new city, you get to enjoy a 15 prod/15 gold boost to go with it. Its really quite strong.