killed my first deer with a 44 in 1968, been shooting them ever since.
Best factory load has worked out to be Remington 240 gr JHPT. Very deep hollow point and expands to about .72 to .74 caliber. Velocity out of my 44 rifles is 1750 to 1800 fps. (Ruger semi and Marlin 1894)
Next best Factory load is either Remington or Winchester 240 gr JSPT. The Winchester 240 gr HPT rates as a soft point as the hollow point is pin sized. The JSP give better penetration and work better on big hogs (300+ LBS), deer go down quick, but not as quick as the Rem. HPT. Speed is the same as the HPT.
Any max load of 2400 or H-110 with any manufactuers 240gr JSP ot JHPT will work. I have at times found Remington bulk 240gr HPTS and bought them in 1000 lots. This seems to be the cheapest route to go.
Keep within the range of the rifle and it will put what ever you shoot down.
If you want around the house varmit loads that are quiet, use 44 mag brass not 44 special, they can cause a deposit ring in your chamber causing feed issues. A mid range load of unique will fill the bill with out much muzzle blast. Try to find the mid range load that hits close to your heavy hunting loads and use a visually diferent bullet so you can tell at a glance which is which.
Above all enjoy your new rifle.
Here is a photo of that first deer