Heavy bolters work very well against Sluggas, you just need one squad to take point and tie them up in melee so that everyone else can shoot. Also note that the Sluggas will be continuously taking morale damage from the meat shield meleeing them back and every now and then killing one of them, enough to effectively zero out their morale regen so that flamers will have a much easier time breaking them. Watch out, though: the UI doesn't show it, but Orks actually get hidden morale bonuses if enough of them are massed together in close proximity.

The Big Mek will be more of an issue due to constant invincibility frames during sync kills, but look at it this way: every second the Big Mek is doing a sync kill against the meat shields is one second it's out of melee range of everyone else. Early-game, send the Force Commander at it. He might not survive, but he doesn't have to: his job is not to kill the Big Mek, but to pin it down for everyone else to shoot.