I bought a Nashbar Hounder single speed bicycle last summer, and I've been looking to do some upgrades....also debating if it's worth upgrading, might be smarter to buy a new bike? Anyways, I wanted to know if I bought bought a 3 speed crankset, a tension wire, and kept my single-speed freewheel, would it work? or would I need to invest in a rear derailleur as well as a 8-9 speed freewheel? Also, I've seen single-speed derailleurs that might work with the 3-front chainring....any Suggestions?

Why not a rear hub like the S-RF3 by Sturmey Archer?

https://www.sturmey-archer.com/products/hubs/cid/3/id/17

get an internally geared rear wheel /thread

You would need some kind of double-wheel chain tensioner to run multiple front chainrings, like Alfine, Sturmey Archer, Paul, etc. An old derailleur will work on the cheap. Basically, you need something to pick up the chain slack when you have it in the small rings.

To do that you would need a front derailleur for shifting, a rear derailleur to take up the chain slack and a triple crank.
Might as well just get a 5 or 6 speed freewheel and a rear derailleur.
Or even better, go with an internal geared hub.

adding an internal gear hub will certainly work but it may not be as simple as "put a new rear wheel on". When I wedged a SRAM S-7 in the back of my fixed gear I had to respace the rear triangle and install a new (wider) bottom bracket to achieve a decent chainline. This isn't a big deal on a steel frame such as yours but it is something to be aware of when deciding between a new bike and upgrades to the current one.

Below are pics of my fixed gear to IGH conversion. This bike is currently back to fixed gear and the IGH wheel just sulks in the corner of my bike room.

SRAM T3 + Capo. Laughs @ hills. Sneers @ head winds.



SS riders have been doing this for a while now.

Ye olde Sturmey Archer AW. I've got one in my main ride as well as seemingly 85% of bikes built between 1950 and 1973. If you are so inclined you could buy an old 3-speed for pocket change, take out the gearhub, and build it into the wheel. They're indestructible and parts are readily available. IIRC they're also almost exactly the same width as a SS freewheel hub.

...and appears to have enough extra cable that it can be installed on a tandem.

/srsly though- looks like fun.

That probably was a tandem cable! I'm shifting that T3 with a low spec Shimano LH (triple) brifter, BTW. Works great.

Sturmey Archer S2C rear hub w/ Sturmey Archer X-FD front hub. 2 gears out back with a brake, no cables necessary. Front drum brake. Pretty much a bolt-on project.

Way back in the early-mid 1980s I had a bike set up with a 3-speed rear and a double-ring crank in front and a rear derailleur for tension. It worked great as a functional 6-speed. IIRC, there was no front derailleur so on-the-fly front shifting was tricky.