I have a very old Mac. How do I transfer photos from it to my new Mac air

The new Mac's Migration Assistant should take care of that. If not, just copy the library to an APFS (GUID) formatted thumb drive and copy that to the new Mac. With a very old macOS it might be best to copy only the user data files anyway (just remember that the hidden ~/Library might contain important info like old local Mail folder).

But if the old macOS is very old, then there might be some obstacles. AFAIR there is limit how old iPhoto libraries the current Photos can import. If that is the case, you could "export originals" (or just "export" if there are pixel or metadata edits you want to preserve) from the old Mac to a thumb drive and import them in the new Mac. You would lose any custom albums, though.

Maybe the best option is to route the images via iCloud Photos so edits and albums are preserved with minimum hassle (maybe with some $$).

Matti's suggestion are great. Another possibility is to connect the two Macs together directly. I haven't done this in years, but others say it works well. See this:

Transfer files between a Mac with Apple silicon and another Mac - Apple Support

(I really need to play with this-- I remember thinking it was kinda cool.)

Using iCloud is also a good idea. Matti says "$$," but really, for 3 bucks you get 200 GB (probably more than the old Mac has) for a month, and then you can unsubscribe to iCloud if you want, afterward. Frankly, it's worth the $3/month to have iCloud Drive and iCloud Photos.

I am Mac / Windows (by force at work). But I can pretty much use either, but not both at my retired age. Total Apple. I must agree with each of you. And I am not sold on beta AI's from anyone thinking for me. Prefer to make my own mistakes. Good site with good suggestions from the Community.

I also use Samsung Pen Drives with both types USB. Old on 1 end and New USB C on the other. Pretty helpful in many instances. I long for new iPhones with USB C, not Firewire. Life must move forward because it is very short.

For those who helped me with the BlueBox Dilemma , I thank you very much. This is one of my plagues. I also think it is related to Intermittent Internet Connections. All the basic software, Messages especially needs to refresh automatically and give us a special key to Force iCloud refresh. They bury Back Up way down there in the Windows centric stacks. Apple Genius Bar did not know this answer and I am 2+ hrs from the store.

I bought this USB drive at a "local" Radio Shack Store at Lake Oconee, Greensboro, GA. Several years ago. Pretty Remote area. I have 2TB of iCloud for my Apples. My 14" M1 MacBook Pro 64Gb/2TB is too much for me. Not many retired people need that. Suggest people budget with the idea of moving up as soon as practical. Not sure Ai is a good idea yet. Causes me a lot of problems that I am spending too much time on. Need a bigger screen ! Much Bigger.

TomE