Lathes...
HUGE topic...
You can go from your basic chinese made model (new, but cheap) to your heavier american/british/european made models (south bend, sheldon, myford, colchester, emco - affordable if they are second hand) to your small precision lathes
check out:
www.lathes.co.uk
and theres more links on my website:
On ye Art and Mysterie of Turning
Materials?
Get them from any reputable steelyard
for a steel gonne go for somthing you can trust like 1055 - mild steel(1020, 1025, 1030, 1035) would be good enough but theres a lot of junk classified under that label coming out of dodgy chinese steelmills so make sure your steel comes off a bar with some kind of marking on it saying exactly what it is.
Bronze - I forget what the US coding is but what you want is around 90% copper 10% tin - "gunmetal" or "navy bronze". We've had a big long post on this before so have a search of past topics.
If you are a begginer I would recommend you start with a bit of thick walled commercial gun barrel for safeties sake. With a vice and a file you can taper the barrel by hand and leave a "cannon mouth" at the muzzle (very HC). Then I'd use a commercial breechplug, remove the tang and braze on a socket to fit to my pole-stock.
If you were machining from scratch, the chief difficulty is drilling a long accuate hole then reaming it smooth. For the outside, you could use square bar, machine off the corners then file the corners flat leaving an octagon profile (using a filing jig on the lathe).
Or you could buy "hollow bar" (precision ground round bar with a hole drilled in it at the mill) and play around with that, but you'd have to fit a breachplug as in my first example.
good luck!