Your bike is probably a one speed with coaster brake with a 26 x 1.75 or 26 x 2.125 inch tire,
so an internally geared hub would use the same chain and sprocket you have now. The new rim should keep the same tire size. I would get the whole same size wheel and just add your tire and tube to it, unless I had experience with lacing up spokes.
To switch to a cluster and derailleur would be much more involved than a internally geared hub. The chain and sprockets are a different size.
They have the coaster brake versions on the internally geared hubs, so you still brake the same way and don't have to add a handbrake. You can go all the way to 8 speeds, but I've found 3 to work fine on a cruiser. There are a good selection of shifters for them.
Older styles or the latest type depending on the look you want. A stick shift on the top bar looks cool, as does an old style twist grip. One of our advertisers here has a good selection of stick shifters you'll see at the top of the page on their ad.
If you decided on a 2 speed kickback, you wouldn't need to add any cables or shifter, they shift by pedaling back slightly.
You have a lower gear and your standard gear like the bike has now with the one speed, so you start off easier and can go up hills easier. The auto versions shifts at one certain speed only, but you choose when to shift on the kickback style hub.
Then pedal backward to stop just like your one speed hub. I've used all the different types except the ones with more than 3 speeds. The 7 speed Nexus is popular too. They have freewheel versions also with no coaster brake, so be sure it's ordered correctly if you get one.
The old Bendix kickbacks, yellow or red band (they have 3 bands on the hub shell) are good too, if you find one in good working order. They were made back in the 60's.
. I haven't plugged the gears into a calculator to see if is worth doing...but, it sounds cool, so it is my plan for now 
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