*WARNING*
Many of these radios were "transformerless", meaning there is no isolation between the service AC and the chassis of the radio. Severe shocks can result with improper handling. The "S-38 Guy" link at the top has info on this I believe. Some units have been modified over the years, or were designed after a certain point and did not have the deficiency.
Darren, unless that unit has been used reliably recently, and/or has had the caps replaced, I would advise caution on simply plugging it in and turning it on.. you could end up with a smoke show, fire or a severe shock. You may want to have somebody familiar with older radios to give it a going over and look for typical problems; of course you may have the ability to do this yourself with some research.
I collect such radios (although I don't current have an S-38E), and after a visual inspection I would normally power them up slowly using a variac to avoid big problems. The old capacitors tend to dry out over the years and become very problematic.
There's likely not much I can tell you about the S-38E that you can't read for yourself with the links you have. Your model was made near the end of a very long run on the S-38; although the appearance is quite different from the early models. Some variants of the S-38E are worth more than others due to scarcity (the beige case is rare to find for instance). These models were more of a "general coverage" type receiver rather than a hard-core ham unit, but many hams and Short Wave Listeners had them in any case (especially the older models).
Now for the antenna connections. Most Halicrafters, including the S-38 had a terminal strip marked A1, A2 and G. Be careful! If the radio has been "monkeyed with" the G screw may have been shorted to the chassis... this is VERY dangerous if the above conditions exist (the power supply issue). Make sure all screws are not connected to the chassis in any way (use a meter to be sure) before proceeding.
Use the A1 screw with a random wire type antenna. You can simply string a wire around a room, or for better reception string it outside between a couple trees (as high as possible). In this scenario you would short the A2 and G screws together (the jumper may actually still be in place) and run a wire from G to a ground connection (plumbing or a ground rod outside).
You can also erect a "balanced" type antenna (dipole,doublet etc) by running the wires for such to the A1 and A2 connections; remove the jumper between A2 and G in this case (but you can still connect to ground). If you can't connect to a decent ground you may be able to get away without it at all and still get reasonable performance.
Enjoy your find! I get a lot of pleasure out of my old radios sometimes, especially in the winter when things are otherwise pretty boring.
. That's fixable with a soldering iron.