Replaced the pump, a pressure valve and the corresponding hose clamps. Plus some misc o-rings, gaskets and such that I figured could use replacing while I was in there.
Just because I didn't know what the problem was exactly, I was going to replace the little water flow sensor which is a mini turbine type thing (water wasn't flowing, so I thought there was a chance the sensor could be the problem) but the new part was assembled poorly. The o-ring was pinched. I assembled the entire machine with the new sensor, fired it up and it didn't work. When I moved it water came out the bottom, so I knew there was a leak. Found the sensor leaking, put the old part back in, reassembled the machine and it worked perfectly.
Total time with takedown, reassembly, partial 2nd teardown and reassembly was no more than two hours of work. Most of the time was waiting for parts. I had to place two separate orders. I didn't discover that the pressure valve was bad until I went to reassemble it after the first order arrived.
Glad I took lots of pictures. I wouldn't have remembered where everything went after waiting a long time for parts.
I think the machine (Gaggia Brera) retails for around $500 new these days. Parts and shipping were under $100. So it's a win.