OK,another opinion!!
Those of us answering you have no idea of the condition of the lathe. If the lathe was working in a commercial shop,running all day,every day,it might be weak.Same for the mill.Few commercial shops would put a Jet machine on the floor. Some schools will.
Or it might have been a home man-cave lathe,that ran 100 hours a year,doing light work.
It might have never been beat on,it might have been brush cleaned,built 5 rifles or a few bicycles or RC cars.
There might be NO REASON whatsoever to do anything but keep it clean and lubed.
Rblevin,I ran a lathe and a mill for 30 years earning a living.If those machines are sound,you have a real prize.
A 13 in lathe will have a 1 3/8 spindle bore.It will accept a 5C collet drawbar .Maybe some day you will get one.Collets are handy.
A 3 jaw chuck is useful.Some are better than others.Its nice if yours has reversible hardened jaws.Don't mix them up!.They are stamped to match the jaws.Some 3-jaws have screws that allow you to adjust out runout.Buck and Yuasa make those,among others.Some chucks are just garbage.There is not anything you can do with them.If you should have or find a nice,value or even used chuck,they attach to a "backing plate" The backing plate is first attached to the spindle without the chuck.Then it is machined true to your machine.It will be faced for squareness,and a diameter will be turned to fit precisely in the back of your chuck body. Then the chuck body is assembled to the backing plate.Things will run pretty darn true.
But likely,you will want a 4-jaw.Each jaw moves independently.With an indicator,you can get your workpiece running true to within .0002 easily.
For gun work,you will want a "Spider" You will make that.Your mill will help.Its a steelcap with a bore that fits over the other end of your spindle.It has 4 opposed screws that you use like the 4 jaw,so both ends of a barrel run true.
MANY smiths are
cursed with smaller lathes that won't allow a barrel to pass through the headstock.They have to have a long bed,steady rest,etc.
You have it easy!!. You don't really need a follower rest or a steady rest for 99.9% of what you might want to do.
Just my opinion,but if you check the barrel making process of makers like Krieger,the stress relief processes,etc,I just do not worry about profiling barrels!!Buy the right profile in the first place.
If you do not have one ,a proper quick change toolpost is ,IMO,a "must have".I like the Aloris type .Small,value priced Chinese knock-offs are available.I think"Phase Two" makes one. You get blocks to hold turning tools,boring bars,part off blades,etc. Its an investment,but a good one.
If you do not have one,you will want a "live center" Some have a set of interchangable tips.Nice,but expensive.A basic one will do.
Some "center drills" A set comes in a round,wood block.Maybe "0" thru #4
You will want to find a "thread guage" on E-bay,also called a "fishtail" Its useful for grinding thread tools,and setting them up in the lathe.
For your mill,look for a "3/8 dia edge finder" .Starret is good.SPI will do.
You will use it ever job!. It magically finds the edge of your workpiece so you can set your dials.
Ita a matter of preference,but IMO,Browne and Sharpe are the best dial calipers. Mititoyo,Starret are good. If you have to go cheap,a good brand VERNIER,no dial,no digits,is cheap and reliable.You just have to learn how to use it!
Get a 4 in Norton fine india stone ,about 1 in by 4 in by 1/4. That will be the stone you use on your lathe tools,and about any other cutting edge.
Get a jewelers loupe.5x,or 7x,or 10 x,no matter. and a good 6 in satin chrome 1/2 wide flexible scale.One side in 1/10th and 1/100th grads,the other in fractions..1/32,64th,etc.
You'll need a 1 in micrometer,too,to get going.
You will need a spindle oil and a ways oil. Texaco Regal R+O is a good spindle oil that is also used in hydraulics.
IIRC Shell Vactra is a good way oil. An outfit like Blue Ridge or Grizzly might sell small qty.
WD 40 is a great aluminum cutting/threading oil. Relton Rapid Tap,Tap-Matic,Tap Magic,etc sell steel tapping fluids and cutting oils.Generally,do not drill or tap metal dry.Except brass.
Feel free to PM me,I will be happy to help.
If you know what to look for,E-bay is your friend.