I am just getting into this sport and do not have a range where I can practice on steel, but will have to practice on paper targets, so I can not use trial and error to determine what loads will topple the targets. I will be loading for 30-30 and .357 in a Contender.

In general, should I work with light for caliber bullets and high velocity, heavy bullets with less velocity, or something in between?

I am not asking about trajectory, just about the power needed to get the job done.

I will be using a scope and barrel length of 12" so I will be limited to a few categories.

Pretty much anything you launch out of a 30-30 will work, power wise. I would try and find the most accurate load with which you can tolerate the recoil for 45 rounds. I would give up a little accuracy to reduce recoil! (I've gone to the extreme of using a .223 launching 77gr bullets with good results.) BTW, I spotted for a new BB UAS shooter using a 30-30 who used factory ammo of two different types pretty much topple everything but the 1/2 size shoot off targets, of which he hit over half!

As for the .357, no personal experience, but folk seem to use somewhere between 158 and 200gr bullets. I think mostly for accuracy reasons. (My usual "slow" round is a .44 Magnum chucking swaged lead 240 gr bullets at ~1000 fps.)

Jonaton is spot on. I used to shoot 30-30 years ago and used 130 grain bullets for the chickens, pigs and turkeys then used 150 grain bullets for rams. Extra bullet weight for rams since they can be hard to tip over. I never went for max loads just something in the middle that was accurate. I figured I had to hit them first before I worried about power. The 357 mag may need the heavier 200 grain bullets since some rams can be stubborn. Even then you don't need to have a max load to get 200 grain bullets to be accurate in a TC. Your two cartridges will serve you well at any match.

SJS, Jonathan and Lynn referenced different loadings and cartridges.
The formula you want is for momentum. I'll simplify for you and use a convenient example.
Since the Ram is the heaviest and most distant we use that animal as the test.
At the Ram: W times V should equal or exceed 168900. Why that number ? It is .75 pounds/feet, and usually serves as a bottom reliable.
The example : A weight of 168.9 gr (think of a .30cal 168 bullet) but all that matters is weight. It strikes the Ram at 1000 fps.
This gives the 168900 we referenced. Any combination of W times V which gives 168900 at the Ram is theoretically workable.
Obviously we aren't going to use a 1gr bullet. A 200gr bullet needs to arrive at 845fps. So a 100gr bullet needs to arrive at 1690fps.
Jonathan's 77gr needs Terminal Vel of 2193 (2200) fps.
The question then becomes------how fast do I need to launch my bullet ? Use JBMballistics.com .
(I am not a mathematician---and I don't play one on TV) But I have rowed this boat since 1980. rp

Thank you very much guys, that is very helpful.

I shot RCBS 200 gr rifle Bullets casted from straight WW with 14 grains of W296 for many years, used it to shoot into international class. Never have played with the 30/30
Good luck, welcome aboard!

Don't see that particular load in my manuals but it sounds like the .357 needs both heavy and fast.

The .357 mag is the marginal cartridge to start with but the 200 grain rifle bullet makes it successful in Silhouette shooting.
You won’t find that load in any manual, it was developed by one of my old shooting partners.

I will give you a different take. The .357 a safe load of a 180-200 grain bullet that is comfortable for you to shoot will be fine. A local shooter uses a 158 through the turkeys and 180 on the rams. The 30-30 again a safe load of a 130-150 bullet that is comfortable will be fine. If you "ring" some rams, safely add more powder or use a heavier bullet. Remember you want to keep your loads under 2400fps at the muzzle. It will minimize target damage and be comfortable on you.