After much research and evaluation of my riding style, I have finally decided on the drivetrain configuration I want. The bike is a mid-90's Trek multitrack.

Crankset: Sugino 46/36/24 (square taper)
Cassette: 12-23 9-speed

I will be using simple friction shifters.

The bike currently has Alivio front and rear derailers. My question is whether the current rear derailer (shifting an 11-28 7-speed cassette) will work with a 12-23 9-speed or if I'll need to buy a new one. I also need to know if I'll need a new front derailer (old one is currently shifting a 42/34/24 crankset).

I'm primarily working on budget figures and I'm not sure whether to include derailers. I've got a couple of bicycle maintenance books but choosing derailers is over my head.

I think I've figured most everything else correctly. I know I'll need a new chain and I plan to let an LBS do the switch so I'll let them worry about getting the correct BB and chainline issues.

Running in friction modes eliminates compatibility issues, so the only serious item is take up capacity for your gear range involving 11t on the cassette and 18t on the chainrings for a total capacity requirement of 29t. 29t, is just slightly in excess of what most road RDs are rated for (Look up the capacity for your RD on the Shimano site to confirm) so you might need one with a longer cage.

But in all likelihood you can skate by without replacing it, if you're willing to never use the granny with the smallest few rear sprockets.

That's the rear, the front may be a different story, if the original chainset was a double with an FD to match, it may not work right with a triple, especially if the outer chainring is much smaller. Also most double FD cages aren't long enough to get low enough for 24t granny. (if the bike was always a triple disregard FD concerns).