Congrats on the new house. My wife and I bought a "new" 1958 home a few years ago and went through the same thing.
While patching the chips in the concrete from the tack strip, I was wishing I had tried cutting off the nails flush to the concrete first. My main concern was matching the color of the concrete. Our slab is colored light-brown-tan (original). Patching the holes was fairly easy, just mixed up some colors with the concrete-patch product and filled them in. Once the concrete dried, we ran a small orbital sander over the patches to smooth them out. Knee pads help make this easier!
Ace Hardware sells a crack filler. Fill the crack, let it dry (it shrinks a little when it dries) then top it off, let dry then sand it level. Use the paint on top to match the color.
As for the linoleum tiles, we had it in the kitchen and two baths. The tiles came up easy and a long scraper took care of the little chips that stuck. The stripping agent for the mastic works well. Just need to pour it on, let it do its work for a while, then scrape it off (get a long pole style scraper with replaceable blades). You may need to repeat in some areas where the mastic is thicker. The fumes are strong, so keep the windows/doors open. Keep them closed if you want a free trip to magic-land to see the sourcerer and his purple unicorn.
It's a bit of work but it looks great and I love having concrete floors. Note: there will be some "shadows" left from the linoleum. I didn't seal the kitchen or bathrooms and over time, due to high traffic, the shadows slowly start to disappear.
Hope this helps,