Initiative is based on chassis (i.e. light/medium/heavy/assault) so if you're fighting an OpFpr with lighter mechs than you, they will generally go first unless you have something that increases you initiative. OFFOR, I believe, also gets to go first in the same phase.
The game is designed such that you will be expected to be fighting 8 or even 12 mechs with four, possibly with turrets as well, because - like literally every other game with AI out there - it has to compensate for, like, not being able to play like a player.
Other than that, without further information, I don't think we can offer much help. Composition of mechs, how skilled your pilots are, what type of mission you are doing and whether or not you are scouting properly all matter, and whether you're using cover etc; all of which are critical. (Also, the assumption is you're playing vanilla, if you're playing with a mod that can also change things quite a lot, especially as the mods are generlly intended to be harder.)
It should be noted that during the early game in particular (in vanilla) you generally need to be doing a good bit of grind until the point you have enough gear to upgrade you mechs from stock and get some decent chassis (because not all chassis are equal and not all stock configurations are particularly good).
Early game, I find if you are struggling, you need to keep to lower skull missions and only try a little higher skull missions on battles (which are straightfoward) and to remember that one of the first things you should do to your mechs is shunt the armour up. Also, especially in the early game, melee is often a better option against lights that low gunnery shooting, as it's somewhat more reliable.
