There are five or six grades of steel commonly available in round, square, and hex bars for turning. Quick guide to the four digit AISI/SAE steel numbering system: First two digits indicate a category for the alloying elements (other than carbon), the last two indicate the carbon content in 1/100%.
1018 steel is a plain carbon steel (10 category) with approximately .18% C. In general, a carbon content less than .24-.25% cannot be quench hardened, must be case hardened by adding carbon to surface. Easily welded but is soft enough to tear when cutting, easy on the tools and can be filed/sanded/polished to a good finish.
1144 steel is often called "Stressproof" and is a resulfurized steel (11 category) with .44% C. The slight sulfur content makes it easy to machine cleanly, the carbon content means it can be quench hardened, and you can get beautiful surface finishes with just a little work. Not recommended for parts that will be fusion welded together.
12L14 steel is a carbon steel that is resulfurized and rephosphorized (12 category) free machining grade of steel (L = lead added) with about .14% carbon. Machines beautifully, very easy on the tooling, but it cannot be quench hardened and usually isn't fusion welded (the lead content causes problems).
4140 steel is the one of the most versatile grades of steel ever developed. It is a chrome moly steel with .38% to .43% C. In the annealed state it machines well, although speeds and feeds have to be reduced compared to 12L14 or 1144. It can be easily quench hardened with a maximum tempered tensile strength of 150,000 to over 200,000 psi. Many blued steel firearms parts are made from 4140 steel.
4140 is also available in a "pre-hardened" state. The heat treating process makes the steel stronger and tougher without making it too hard to machine. It can take on a good finish but speeds and feeds must really be reduced. Even with carbide tooling it takes a while to machine compared to the free machining grades.
4140 can be fusion welded but you need to pre- and post-heat the parts or you will get a brittle weld that will fail. 1018 can be welded cold.
8620 steel is a nickel chrome moly steel with .2% C. It is strong yet easy to machine; it was designed specifically for case hardening purposes. Case hardening makes for a very hard surface with a softer and more ductile inner core.
A lot of plate, tubing, angle, and other wrought shapes are available in A36 grade (falls within a different steel numbering system) which is about the same as 1018. There are other grades of structural steel available but in my experience its often a special order thing.
In stainless steel there are usually only two grades available in rod stock - 304 and 316. 304L and 316L are designed to be weldable grades of 304 and 316 stainless. The same material is also available in angle, strip stock, etc. All SS is a pain to machine. Slow things down, use a positive feed with sharp tools, don't let your tools rub the surface. Cut or get off of it, dull tools and high pressure can work harden the surface and make it almost impossible to cut through.
In aluminum the most widely available grade is 6061. It is easy to machine but is not one of the stronger grades of Al. You can also get 2024 and 7075 in rods, bars, and plates. These last two are "aircraft" grade high strength Al alloys.
The steel rod you get in 2'-3' lengths at the hardware store is probably 1018 or close to it.
I've used bolts as raw material for turned parts with some success. I've had nothing but grief trying to use rebar as raw material for lathe projects. Often has hard spots, inclusions, etc., I think they must recycle a lot of junk steel into rebar.
I've also used old chrome plated cylinder rods (typically 1050 steel) but the problem is getting under the chrome. I have chucked up the rod in the lathe and used a cutoff grinder to cut a groove around the shaft, then switched to a regular cutoff tool to finish the job. Once you can get to one end you can turn down the OD to remove the chrome plating, although you will probably cut a notch in your tool where it cuts off the chrome. 1050 machines nicely, finishes well, and can be hardened by quenching.
Some used parts from cars and other equipment can be used as raw material, but I would stay away from anything that looks like it was hardened. If a file won't cut it your HSS tooling probably won't either.