Upgrading crankset on mountain bike - what are my options
Probably you have already figured out an answer from the given comments, but here is my answer.

I have a bike with a very similar set up. It is an old MTB with rigid fork that I turned into a fast commuter. Like you, the first thing I did was changing the tires to a pair of "1.3 Continental Sport Contact. Later, I changed the crankset and the cassette, which happens to be the same as yours.

I am using a triple crankset Shimano FC-M171 (48-38-28). You can buy it new for around 20€.

About the required adaptations, I would say:


Don't change the front derailleur, probably yours will have enough room. You will have to move it up a bit. A rule of thumb is to leave 2 mm clearance between the bottom of the outer cage plate and the teeth of the large chain wheel, but you'll have to try to find the best fit. Sheldon Brown has a nice guide about derailleur adjustment.
Replace the chain, which is advisable when changing the crankset and/or the cassette (they all wear out at the same time). Also, the new crankset will require a longer chain. I'd say something between 112 and 118 links. It is preferable to buy it longer and then remove the links in excess.
Probably you will need a wider bottom bracket (larger axle length). This is a requirement for the new crankset, otherwise it may happen that the smaller chainring will touch the frame. E.g., in the case of the FC-M171, it requires an axle length of 122 mm. The diameter of the bottom bracket will be the same of your current one. A low-end Shimano squared bottom bracket (sealed) costs around 10€ new.


I must add that for me this amplitude is great for every occasion, I can climb reasonably well and I can also reach 50 kph with a reasonable cadence (not sure, probably something between 80 and 110 rpm).

Go for it.