I would not replace the gearbox, hop-up unit, or the barrel unless they are unusable.
For accuracy, the cheapest thing you can do is change the hop-up bucking and hop-up nub. I don't know what it the most popular brand right now, but I've always had good results with MadBull Red buckings. There are a couple popular hop-up modifications you can look up, one being "flat-hop", the other being "r-hop", if you do them right, you should be able to get some good range out of your gun. Another really important thing is to start using heavier bbs like .28g. They will increase your accuracy tenfold and travel further.
For rate of fire and trigger response, there's a lot of room for improvement, even when using NiMh batteries. Here's hoping that you at least have limited knowledge on how to work on wiring... Replacing the rectangular Tamiya-style connectors with bullet connectors or Deans-style connectors on batteries and your gun will dramatically improve rate of fire. The science here being that a current will only have as much voltage (speed) as the the most resistant part of the circuit will allow. Since Tamiya-style connectors have very tiny prongs, this slows down the current to the highest speed the connectors will allow. Subsequently, replacing any thin-gauge wires, and using batteries with thick-gauge wires will also help. If you want to use LiPo batteries, go to an online hobby store and get yourself a charger and some 7.4v batteries, and maybe 11.1v if you feel the need. They will be far cheaper than an airsoft store. You'll have to replace the connectors on your gun though.