You are allowed to tech any card you like at any time. You could tech Tech 3 cards on turn 1 if you really wanted.
Tech buildings just say what you can and can't play normally. There are cards which allow you to play things you're not teched into, and they'll say that on the card.
For example, this card tells you to "... put a unit of the same tech level and the same cost or less from your codex into play. (Even if you don't meet the tech requirements for it.)"
Admittedly that pulls directly from your Codex. Let's find a better example....
Feral Strike allows you to "... put up to two units from your hand into play if you have tech buildings of the same tech level as them."
So that doesn't require you to have be teched into a spec, you can just put any spec unit into play.
One more important thing: Spells don't require Tech buildings, but they do require the hero of that spec to be in play.
So for both of those cards I just posted, you don't need a Feral Tech 2 or anything, you just need a Geiger or a Calamandra respectively. (Admittedly, the Calamandra must be maxband at the start of the turn to cast an ultimate, but that's slightly unrelated at the moment.)
Hope that clears some things up.