The TCW hub is a series of compromises. It is basically a coaster brake grafted onto a modified AW drive. Some parts interchange with the AW, but many do not. It's not exactly an AW, and it's not exactly just a plain coaster brake, but a combination of the two. A bicycle with a TCW hub should always have some other form of brake, preferably a good, front wheel handbrake.
If you have an unserviced TCW hub, you'll want to open up the coaster brake and clean and re-grease with a quality grease. It need not be expensive bicycle shop grease, but it should be something better than the old, thick axle grease that drags. For light duty use, I'd use something like Lucas green grease (hardware stores have it), or Lucas red high temp grease if you're really going to be heating up the brake. I don't see just oiling the hub as an option here because the coaster brake side of the house really needs a decent quality grease. The brake side bearings should get a good coat of something like the Lucas green grease as well. Marine grease is also an option if you are going to be riding through standing water, but that's generally not a great idea.
The transmission side of things is a little bit like an AW hub - grease the outer bearings with something like Lucas green or another quality grease (again, not the old, gooey axle grease that drags). The innards get a coat of light or medium oil - 20 or 30 weight is fine. Adjust the drive-side cone to correct tension, then set up the coaster brake side cone for play.
When using, be careful not to slam the shifter down into high gear while also braking or transitioning to braking at the pedals. The reason for this is that the two halves of the hub are held in place by a single, fragile E-shaped clip that sits in a groove in the axle. If that E-clip is forced out of the groove, the hub will cease functioning correctly and you may fall into neutral, preventing the hub from driving and preventing it from braking.
One other thing to remember is that by modern standards, the TCW's brake ring/shoe is undersized, so braking power is somewhat reduced. The braking mechanism is also applied AFTER gear reduction/increase through the transmission, so you will have somewhat reduced braking power in high gear. You will have NO braking power in neutral between normal and high.
The TCW version with small, curved arm and the traditional writing on the arm is the older version of the hub.