The Florida East Coast yard along Phillips Highway (US1) is a good bet. The entrance is near the intersection of Phillips and J. Turner Butler Blvd. (Hwy. 202), across the street from Starbucks (IIRC). I had heard that they tolerate railfans as long as they stay in the yard office parking lot, and found that to be the case when I went there last June--I went there twice, and nobody said a word to me. From the parking lot, you can see both yard and diesel shop action. A nice place for FEC mainline pictures is just south of the yard, where Sunbeam Road crosses the tracks, just off Phillips. (Of course, catching any FEC mainline action is somewhat problematical.)
A good spot to watch CSX is at "Grand Crossing," where CSX and NS cross, at the dead end of Kings Road North on the west side of the tracks. (Note that Kings Road exists on both sides of the tracks, but does not cross them. WARNING--this dead end street is very secluded (trees and brush) and is a perfect spot for doing drug deals and dumping bodies, so be very careful if you go there, especially alone. Just west of there, the NS yard lies along Kings Road, but not much seems to go on there. There's nothing to see at the Amtrak station at all, unless there's a train there. There's a switching road in the port area called the Talleyrand Terminal RR, which is a Genessee & Wyoming property. Their office and yard are located where the tracks cross Talleyrand Ave. I checked with the guys in the office, and they didn't mind me taking pictures. The scenic highlight is the bridge downtown that carries FEC and NS trains over the St. Johns River (although I didn't have time to get there, so I can't give a first-person description). The railroad bridge parallels the Acostia Expy. bridge, and it looks like you can see it from Riverside Ave. on the north bank or Prudential Drive on the south. If you look at railpictures.net, pick "Florida" and use "Jacksonville" as a keyword, you'll see quite a few photos of it. Hope you see this in time (it's 215am in CA). Good luck, and let us know how you do.