Like most, I beeline key techs for a variety reasons - it could be for a key wonder or building that I need to build, new unit I like, possible change of government, key resource I'd like to take advantage of, or the 'first-to' bonus.
With all of this included, several techs stand out. When considering the 'first-to' bonus as the priority, I really like to head for Monarchy for the early great person - the first of only two that offers this. A great person at this stage can really set things up, but the added bonus of being able to utilize dye is great for any science or gold city you are trying to develop for sure. It also allows for the Monarchy government of course - nice if you'd like to push some culture in your capital to speed along another great person (though I still like to remain in Republic early if I am trying to REX). This is another reason to get peeved when you are up against the English - no Monarchy bonus for you!
Invention is similar. Another key person that you get (and deny the AI) and opens up a key wonder in Leo's Workshop. It also leads the way to the key techs you'll need to eventually beeline combustion.
Other nice techs with great 'first-to' bonuses that also offer other good reasons to beeline include Currency (before breaking 250 gold will get you banking at that mark of course), Industrialization (the needed factories), and Communism (opens up the production-rich government).
I see a lot like the irrigation bonus, but I rarely ever get this. Although I like to expand fast, I usually don't have THAT many cities to really take full advantage of this bonus. Further, unless I have wheat in a few cities it does not offer too much beyond that. I think there are more important and rewarding techs around this era to go for. Mass Media - appearing later in the game when I am likely to have a very good number of cities - is a great one to hit.
There is actually one that is not on the list above that I like to get early and gives a nice bonus as well - Navigation. Assuming the Spanish are not in the game, getting to Navigation first gives a free Galleon, and that can be very nice to have for help in defense, city attacking, quick settling and exploring of islands, or Atlantis-hunting. Nice not to have to build one outright. Also, allows you to work whales which can be helpful depending where your cities are located.