I'd suggest starting with 6, see if you like the general premise and then play 4 and 4 answer after that, and finally I'd suggest the 3 series after that, the few games based on 3 and including 3.
I really wouldn't suggest the first two games unless you absolutely fall in love in the series and can't get enough of it. This would be my suggested order as someone who played armoured core 1 on release and played every game since then. 6, 4 and 4A are the best games in the series IMO, but the 3 series is the best of the older games, I couldn't play 2 last time tried it, controls are abysmal.
It also depends on what you want from a mech game, I f you want heavy, stiff clunkers that is deliberately so by design, the first three excel at this, not so much 4 or 6. 5 is very much an in between game, they tried to course correct from fan backlash to 4, and 6 perfected the formula.
I got all endings in 6 and it was great, is now my favourite in the series.