I grew up with an 1861 Colt Navy Dad brought home from an OGCA show in the late 60's. It was my job to disassemble, clean and reassemble after a day at the range & reassembling a colt Cap & Ball revolver is a skill you'll never forget. I have that gun today & it swtill looks & works like new.
I'd always wanted an 1858 Remington & thru a mistake from Cabelas, ended up with both .44 and .36. Remingtons are easier to disassemble & reassemble & sighting is more conventional than the notch-in-the-hammer Colt sights. Nevertheless, as others may have already stated, these cap & ball pistols are more accurate than modern revolvers - it's true.
I have an 1860 Army that just happens to fit an old shoulder stock & accuracy is outstanding. My favorite? Whichever one I'm shooting at the moment. My latest is an 1851 Colt Navy, but compared to the 1861 Navy, the 1851 is second best of the .36 cal's.
I have a 2nd model Dragoon, which is fun to shoot, but a lighter revolver gets more range time due to shooting comfort of less weight.
Uberti vs Pietta doesn't really matter anymore, since they both use state of the art manufacturing. .36 is a little higher velocity than .44 for target loads, but both are accurate.
My recommendation is to buy by price. Near major holidays, Cabelas often has sales, including either $5 shipping or FREE shipping. If both are cheapest, get the Remington, since it is easier to disassemble & reassemble. You'll likely accumulate several Colts & Remingtons whenever they are on sale.
A benchtop cylinder loader is a handy accessory, especially one you can switch from .44 to .36. If you want accuracy, 33 gr 3F with a paper or felt overpowder wad for any .44 cal can't be beat. 17 gr 3F for any .36 cal. I've found to be optimum. If you need hotter, load it up to the max & blast away, your will lose accuracy. If you need to hotrod it with conicals and some of the BB substitutes, go buy a .45 colt or .44 mag. There's nothing a conical can kill that a roundball won't kill as easily.