It's worse on Cripppling and Maso because the AIs get bonuses to Influence on top of others. On Maso, once the AI grabs a planet in a sector, the entire sector is theirs.
The higher the difficulty the further you will be left behind on tech after the colony rush. Trade for the critical techs that you need from the minors first (because they tend to possess a lower diplomacy level than you and thus trades are cheaper), the get them from the majors. It's even easier and cheaper if you take diplomacy picks at startup.
You can safely skip some of the useless starting techs by trading for more advanced techs. For example, if you are a rusher, trade for Energy Channeling and Space Militarisation and voila, you can immediately research the killer Interstellar Tactics, or if you like Turbo Phasers, for the frigates and cruisers.
Other useful, cheap and early techs to grab from the AI are phasers, shields, trade, advanced trade, basic environment control, industrial theory, controlled gravity, etc.