How much do you want to spend? How much power do you want? More power=more money. Whats it going in? Stick or auto?
You can do a cheap rebuild of just rings, bearings and gaskets. But that wouldn't be much of an upgrade. To do it right you should compleatly disassemble it and take the block, crank and heads to the machine shop to have them degreased, magnafluxed and checked for truness. The block will probably need to be bored and maybe decked. The crank may need to be turned ($70 around here). If the crank is to far beaten down to be turned you could upgrade to a 383. The cost is about the same.
If the heads need new valves, or you're thinking of putting larger ones in. It may be cheaper in the long run to buy an aftermarket set. Even the worst set of aftermaket heads will flow better than a set of factory ones.
The cam you choose should match your intake choice. Duel plane is better for low/mid range. Single is best for upper rpm's. A high lift/long duration cam would be best paired with a single plane and a higher stall converter if you plan on using an auto. Might even need to regear the axles to keep the motor in the happy range.
If you're planning on big power (600hp or more) a factory block will be at it's limits. From what I've heard/read for a good foundation use a four bolt or better yet a 2 bolt converted to 4.
I've built 5 350's all with differn't power levels and for differn't vehicles. The cheapest rebuild (just under $600) didn't last that long. The most expensive one (over $4500) is still going strong. And that sees mostly drag strip time in my Camaro.