What’s the best way to get 35mm film photos onto a computer at high quality?

These days, there are basically five ways:

1. Get the film developed, and scan the negatives. (Or develop yourself, depending on enthusiasm, space, and time for that — easier with black & white, hard with color.) Unfortunately, this was easier 20 years ago than it is now, since there aren't as many scanners made, especially for film other than 35mm — Nikon for example no longer makes their scanner, but I was actually a little surprised to find that there are quite a few options still. See this 2016 guide from B&H photo.

2. Get the pictures printed and scan the prints. This is easy, and can be done with a cheap consumer scanner — or more likely, an "all in one" printer with a scanner built in. Seems like this is most consumer printers these days anyway — they need it for the copy function, so might as well have a scanner. The advantage here is that you probably already have everything you need.

3. Use a DSLR as a scanner — either with a light-table and the negatives, or just with the prints. The latter requires a little bit of careful setup to get good lighting without glare (get matte finish on your prints!), but can work pretty well. PetaPixel argues that it's better than the flatbed-scanner option.

4. Have your film developed at a lab that offers a "develop and scan" service. Most places do this. The general drawback is that they'll generally charge a lot of money for this, and the default may be ridiculously low resolution, with an upcharge for higher-res, and you'll have little control over the results. However, this is hard to beat for convenience, especially if you're already having the film developed. (I shoot film occasionally mostly for the fun of it, and this is usually what I do.)

5. Send to a scanning service like ScanCafe (not an endorsement, just an example). You'll get good quality for low cost — but of course the turnaround is longer.