The bullet on the left is a 130 gr copper Barnes bullet, on the right is a 180 gr Remington Corelokt, both are 30 caliber. Generally speaking the rule of thumb is to drop down at least 1 and probably 2 bullet weights. And copper bullets work best if you can get them to leave the muzzle as close to 3000 fps as possible. I'd drop down in weight until I can get 3000 fps at the muzzle. You'll get just as much penetration from a 130 gr copper bullet as a 180 gr lead bullet. But if the copper bullets impact at 2000 fps or less you'll get very little expansion. This is the reason for the need for speed.
Just to add, this advice pertains mainly to bullets 30 caliber and smaller. The larger caliber bullets seem to expand better at slower speeds. The newer TTSX's with the plastic tip expand better than the older TSX without the plastic tip.
Copper isn't as good for long range work. When you reach the point down range where speed drops below 2000 fps or so you are at about the max range for that bullet. Many softer bullets will still expand at 1600 fps and will work better down range.
Another way to look at it. A copper bullets will retain 100% of it's weight, or darn near that, after impact. Most cup and core bullets retain at best 50%, the bonded bullets around 70-80%. I can shoot copper 130's at 3000 fps from my 308, or 180 gr lead bullets at 3000 fps from a 300 WM. After impact the lead bullet from the 300 will weigh somewhere between 90-140 gr. The 130 gr copper bullet will still weigh 130 gr and come close to if not penetrate deeper than the 300.
Unless hunting dangerous game such as brown bear I'd not suggest anything heavier than 150 gr in 30 caliber. Lots of guys have been using 110-130 gr bullets in 308 and 30-06 with great success. Most of the complaints about copper bullets failing is when guys shoot the same weights they have been using in lead and they impact too slow to give good expansion. The game usually dies, but often after traveling farther than most guys like.