It all really depends on a number of factors, including how well you're doing in lane and who you're playing against. Sometimes just buying part of an item is better than completing it right away (
), whereas sometimes you want to complete something as soon as possible (
). Your first item will also likely vary depending on the above factors, which may or may not put you behind.
Generally though, you want completed items as soon as you can get them. The unique passives/actives of completed items make them far more valuable than just their stats would suggest, so choosing one item and trying to complete it as quickly as possible is usually a viable approach. I would however suggest that you make a point to get tier one boots early on the game, as the bonus movespeed makes returning to lane quicker and can help you survive otherwise lethal ganks. Upgrading boots from there depends on the role, but can be put off for a while depending on how well you're doing.
So, to tl;dr myself: yes, "rushing" one item first is a good idea, provided you threw boots in there somewhere.