Cheers Mark, welcome to Bike Forums!
Any bike can be used for commuting. But the more you ride, the more you'll appreciate a reliable quality bike. Regarding what you have now: straightening the rims is called truing and it's something you can do yourself. Googling bike wheel truing instructions will give you a lot of reference on how to do it. However, if the rims are severely out of true, or if you have a lot of broken spokes, you may have to replace the rim(s) and/or wheels (or rebuild them). At that point it may make more sense to start searching for a used bike in good condition instead of throwing money at your old bike.
As a commuter, it's good to know how to fix certain basic things in your bike. Flat tyres, broken spokes, replacing brake pads and such. Knowing how to adjust brakes and shifting will make your commuting more enjoyable. Bikes are simple machines, so it's quite feasible to even do all the maintenance yourself, if you're so inclined. Some special tools are required, but it's not a huge investment.
As to hybrid vs. road vs. something completely different... nobody can really answer that for you. If you can get your old bike to rideable condition, test ride your commute for a couple of times, and you'll have an idea of what you need. Some things to consider:
- How much stuff will you be carrying? Change of clothes, lunch, laptop, maybe room for groceries on your way back home? Some people prefer panniers, others simply carry a backpack. I prefer panniers myself, but they require a rack. You can fit a rack in most bikes, but it's easier if the bike has the necessary mounting points ready. Hybrids usually have those.
- Weather. Hybrid / city / MTB bikes will give you more room for full fenders, mudflaps and even studded tyres should you need them. I do.
- Road conditions. Again, hybrid etc. bikes give you more options on tyres (width, puncture protection etc) because they have more tyre clearance.
- You will need some accessories, such as lock, helmet, lights, riding clothes. What kind exactly again depends on your preference and local conditions.
--J