By big Bugs I assume you mean Impaler, Tank, and Behemoth?
Ok, so if you are playing without the DLC then the RX-1 Railgun should be your go-to weapon. When fully upgraded it can stun ALL Bugs for a couple of seconds in one shot (except for the smallest patrol units - they die from anything in one hit), keeping them in place long enough for you or your friends to do anything you need to do - retreat, do objectives, reload, throw Stratagems, etc…
The Railgun will stop charging Tanks and Behemoths in their tracks, allowing you or your teammates to get out of the way and avoid death by getting run over!
The stun also forces Impalers to retract their deadly tentacles, which is good if they are about to destroy an objective or kill your teammates, but also bad if you actually want it to expose its weak spot.
Please note that the Railgun is very ammo-hungry and I recommend you DON’T use it against smaller Bugs (such as the Scouts and Vanguards). Use your sidearm for those! Even if not fully upgraded (although that is recommended) you need nothing else other than your trusty P-2 Peacemaker pistol to deal with the smallest Bugs. Alternatively you can bring an MG-94 Machine Gun to deal with those, as well as the various Bug mobs.
If you are playing with a full team then it is usually enough if 1 or 2 people (at the most) wield the Railgun into battle.
As for actually killing the big units, the best NON-DLC support weapons to kill the big bugs with are the RL-112 (best used with a reload buddy ) or EAT-17. Please note that the Bug Behemoths can sometimes survive a direct shot from either rocket launcher, requiring a follow-up shot. As for the smaller Bug Tanks and the Impalers, you can use pretty much any weapon to shoot their fleshy weak spot (Tank Bug has its weak spot on the back, Impaler in the front, but only exposed when it burrows its tentacles into the ground).
If you have the DLC, then the TOX-13 Avenger is, hands down, the only weapon that you need to deal with any and all Bugs (but bonus points for having a teammate bring a FLAM-40 Incinerator and playing Bug Exterminators with you ).
Another fan-favourite DLC weapon against the Bugs is the M-25 Rumbler. Although it is a slow, clunky, cumbersome and volatile weapon (beware of accidentals!!!) it is also versatile enough to deal with anything in the entire game - there is nothing it can’t kill, but that includes yourself and your friends more often than not!