I am going to be shooting a fallow in January.
I am shooting a .243 break open, i don't know the brand name or model.
Mainly i am just wondering what kind of ammo would be best for target accuracy as well as what shooting accessories would help to make it have better grouping.

Depends on twist rate. But a 100 gr. Sierra Game King spire point boat tail will hit where you want it to and perform well on impact.

Add a Harris BRM-S 6-9" to the rifle, a good rear bag, and shoot from the ground to squeeze accuracy out while zeroing and testing for different distances.

Sounds like a TC rifle....

anyway


You can make shooting bags out of the legs of spandex or sweat pants. Zip tie one end closed. Fill one end with bird seed, or play sand. Zip tie the other end closed.

Stress balls also work for under the buttstock. Simply squeeze to raise the rear. Loosen your grip to lower. Panty hose with birdseed also works.

If you can find them. Trailer houses have jack stand looking braces. Some have as low as 6" tall base. They make good shooting stands. Usually you can find a couple in old parks that had a trailer moved.

Old school tripods for cameras can be used. Goodwill usually has one or two.

Take some bino's with you. Or you will be walking back and forth.

Get some shoot N' see targets. They help alot.

You can print targets out. All kinds from animals to red dot zero in jobs. Just google targets. Most come in PDF format.

And the one thing that helps the most if you will be zeroing out in the woods....Bug spray.

It all depends on what your gun likes. Hit the range as often as you can and try different brands/types of ammo until you find what shoots the best. .243's are generally not real picky, but you want the best accuracy you can get with a decent quality bullet. There aren't a lot of mods that you can make to a break-action rifle, but a lot depends on the situation. If hunting in the open, the bipod will help, if hunting from a blind, not so much. The best accessory that I can recommend is good glass.