Silverbird:
I never think it's a good idea to paint over nice wood.
The problem is that the amount of work involved in restaining is enormous. You're going to have to strip all the old varnish off, and then sand or plane the surface of the wood down on both sides to remove the stained wood, and then stain and apply a new clear coat.
What I'd be most inclined to do instead is to simply spruce up what you have. If it was me, I would:
1. Use mineral spirits to clean the cabinets and cupboards first. Removing years of old cooking grease will make a big difference in the appearance of the cabinets right off the bat. Often, the amount of cooking oil residue on the cupboards is so thick that you're thinking that the mineral spirits are dissolving the varnish. That won't happen; varnish (like all drying oils) won't be affected at all by mineral spirits. The brownish crud you're removing is not varnish, it's dirty cooking oil.
2. If there were any smokers in your family, first use water and a detergent (Mr. Clean is actually an excellent cleaner, along with Fantastik) to remove the cigarette smoke residue that may be on the varnish. Then, after removing any cigarette smoke, see if there's any nicotine staining of the underlying varnish by wiping the cabinets and cupboards down with a dilute bleach solution (1 part bleach to anywhere from 4 to 10 parts water).
3. There may also be natural yellowing of the varnish on the wood, making it look deteriorated. Most people are unaware of it, but this yellowing is completely reversible. By exposing the yellowed oil based paint or varnish to direct or indirect sunlight, any yellowing of that old varnish will disappear over a few weeks. The kitchen looks like its on the shaded side of the house. You would do well by opening the blind and positioning a mirror to reflect the Sun's light onto a white surface inside your kitchen to reflect that light around your kitchen. Within a few weeks, any yellowing of the old varnish will have disappeared. (The curators of museums do this all the time. Any paintings over 50 years old were done in oil, and so it's normal to remove the yellowing that occurs in storage before putting these paintings on display. And, that is done by placing the paintings in a well naturally lit room and allowing the sunlight to reflect of the walls onto the paint. Direct sunlight would remove the yellowing faster, but might also fade the paint and fading is irreversible.)
Since the house was built in 1960, then they would have used real varnish over the stained wood, not polyurethane. That means that if you clean those cabinets and cupboards with TSP, it will etch the gloss, and after rinsing with clean water to remove the TSP residue and allowing time to dry, you can wipe them down with a rag dampened with wiping polyurethane to restore the original gloss to those cabinets and cupboards. Keep the rag tightly sealed in a plastic bag in your freezer when it's not being used to prevent the wiping polyurethane in the rag from curing and hardening up. Polyurethane will get thick when it's very cold, but it won't freeze and won't be harmed by the cold temperatures. Wiping polyurethane uses smaller resins that dissolve more readily into mineral spirits, and so wiping polyurethane is much thinner (less viscous) than the stuff you brush on. That allows it to self level very rapidly so you don't have "brush strokes" left behind by the rag.
I believe that if you can restore the original appearance of those cabinets to something similar to what they were like 50 years ago, you'll be happy to keep them just as they are.
PS:
I don't like the idea of painting, but if that's the route you choose, then the best way to do it would be to clean with TSP first to etch the gloss of your varnish, then prime with an INTERIOR alkyd primer (and pay extra to get a high hiding alkyd primer) and then paint with an INTERIOR alkyd paint. For the working surfaces, like the shelf tops, you want the hardest user friendly paint you can find, and that would be either a urethane fortified alkyd paint (like Benjamin Moore "Melamine" in the 303-90 tint base) or a full blown polyurethane floor paint (which is essentially hardwood floor finish, but with pigments in it to give it colour and opacity). The problem is, you're likely to have trouble finding a polyurethane floor paint in a lighter colour cuz that's simply impractical for a floor. Typically, floor paints come in pretinted colour (for the fastest drying time) and you have your choice of only 4 or 5 colours; typically light blue, dark blue, light grey, dark grey and reddish brown. I use the BM product mentioned above and tint it with 12 drops yellow oxide per gallon to make a light yellow (which helps to hide the natural yellowing that occurs inside closed cupboards) and I find that paint stands up very well.