What is the best way to increase gear inches on an old bike?

Question

I only recently got into bicycling and without knowing anything, I bought a bike from a local used bike distributor.
Here's what I've learned about the bike.

• The frame, seat, and handles are original to the bike which is a Univega Alpina Uno.
• Somewhere along its life it had the wheels changed, and the cassette and derailleur were changed to go with it.
• The chain ring is 48/28 (48,38,28)
• The Cassette is 14/34 (14,17,20,24,28,32) Edited to indicate I had incorrectly counted my teeth on the smallest sprocket/cog thingy
• The Front rim is 26X1.75 and the back rim is is 26X1.5 but it came with 26X1.75 tires.
• The derailleur says suntour XC Sport 7000.
• The BCD is 110mm and the crank is 170mm.
• A friend looked at it and said "oh look you have a free-wheel" which didn't mean anything to me but seemed a relevant bit to pass along. My chainset says "Computer designed Bio Pace" on it which I think means it isn't a circle.

OK that's all the information I think I can provide.
Now for the problem.
In the stiffest gear on flat ground I am out-pedaling the bike. I don't know if there is an official term, someone said I was "topping out" but basically from what I gather my gear inches is not high enough. Articles I've read indicate that 11 to 50 from cassette to chainring is a good number to have, and I'm at 14 and 48.Edited to change 17 to 14 since I can't count
So what is the solution?
As I understand it I need to go smaller on the back and/or bigger on the front. But I haven't been able to figure out what is involved.
It seems like if I replace my crankset and my cassette I'll get what I want, but that is a lot of parts and will require a new front and real derailleur.
I learned that the chainring can also be replaced independent of the rest and might be easier, but 48 is already pretty close to the 50,52 that I'm seeing as more common. So it might not be helpful.
I don't want to buy a whole new bike worth of parts, but it is starting to feel like that is what I am needing to do. Is there something I missed?
Edited to add pictures; Here is a google album of the pictures I've taken to help illustrate the situation. Hopefully they match up to the description I've given so far. https://photos.app.goo.gl/CPcJu4fdb8Z2i39J8
Edited Again
I used my calipers to measure the space between the inside mounting points of the frame and I came up with 126mm. The outside measures 139mm. But from what I've read common sizes are 130mm and up for the distance on the inside of the fork(?) so 126 is odd, but it would explain the extra spacers I see on the threaded shaft that goes through it.

Answer 1

A common phrase for fast pedalling is "spinning out" where you start bouncing in the saddle because your legs are rotating so fast. That speed varies between riders but could be anywhere from 100-120 RPM on the cranks.

For "fast" or Pro riders they'd aim to pedal at about 90 RPM. The song "The Gambler" by Kenny Rogers is around 87 BPM so riding in time to that gives an idea of how quickly the feet have to rotate.

For new riders, they might mash along at under 60 RPM.

You should aim to pedal at whatever cadence you find comfortable. Sometimes called the self-selected cadence and its different for everyone.

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Using https://www.sheldonbrown.com/gear-calc.html we can input values as given, (using 1.9" tyre because 1.75" is not available) and get a results table that says you're doing 27 km/h in 48:17 gearing at 80 RPM.

If you can get a freewheel that has a 14 tooth smallest gear, then that increases to almost 34 km/h in 48:14. This is probably the first thing to try. You'll need to fit a new chain at the same time.
14-28 is a common size for a replacement freewheel and 14 tooth is also about the smallest size you're likely to find available new.

To get MORE speed at the same RPM you could change to a cassette, which lets you get down to an 11 or 12 tooth gear. With the same chainring that gives you 43 or 40 km/h but realistically on a MTB you're going to be hitting the air more and resistance will prevent those kinds of speeds without a tailwind or downhill.

If you changed the big chainring, there are risks of it hitting the frame. Check the right-hand chainstay and see how much clearance there is with the existing 48 tooth. Going from 48T to 50T or 53T chainring gives you 29 and 31 km/h at 80 RPM with your 17 tooth rear gear, so much less of an increase. Do both cassette and chainring changes and you'd get 43/45/47.5 km/h from an 11 tooth cog, if you can push it.

Assumptions:

• The wheel/hub interface is a standard threaded mount. This is 99% likely.
• You'll need a special freewheel removal tool, which are cheap. And a big spanner or a bench vise.
• Some level of mechanical competence to make these changes
• Chain-tool to shorten and join the new chain.

Personally I'd go with a 14 tooth capable freewheel and ride it for a while. 3 fewer teeth in the rear gives more difference than 5 more teeth in the chainring.

A cassette swap would require a replacement back wheel, which are not rare for 26" but there's a lot of additional changes and it can get expensive fast. Keep your eyes open for a cassette rear wheel to give you options in the future, which may also give you additional gears but may require a new shifter too.

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Update: OP's freewheel has a 14 tooth small gear, and that rules out a replacement freewheel as being helpful.

14T is about the minimum size because of mechanical limits. I heard there were 13T in the past, but can't find any now, and a single tooth isn't going to make much difference.

So your two remaining options are to put a bigger chainring on as long as it clears the chainstay, and/or replace the rear hub so it takes a cassette and then fit a cassette to suit.

A hub replacement is a lot of work so look out for a replacement 26" wheel, or a complete donor bike. The best option would be a 8/9/10 speed cassette, and a flat-bar shifter to suit your handlebars.

You can also use a 7 speed cassette (yes they exist) with a spacer between cassette and hub, to get the lockring cinching down.