I see your immediate priorities as:
- Saving money
- Roadside repair
Which are all interlinked: a problem you can't fix can cost you a lot in getting home; using your commute and errands as training time saves petrol or public transport fares (I assume you've optimised this)
So immediate purchases should be
- a pump and puncture kit
- a multitool
- a decent lock - you can't afford to lose your bike
Each of these lines should be doable within your weekly budget. I suggest this sequence if you're happy to only take the bike places where it's secure without a lock. Other early purchases:
- Chain oil (the cheapest in a bike shop)
- Some lights
- A couple of spare tubes
Something to do soon is investigate whether there's anywhere you can use tools for free or a nominal cost: a bike co-op, bike cafe, employer's facilities etc. This will affect your future purchases. Similarly if you ride to work, you get to know the other cyclists - many (not all) will lend you tools, help with repairs etc.
Keeping the chain and gears clean and thoroughly but lightly oiled will make them last longer and save you money.
It sounds like you want to maintain your bike yourself. On a budget that means buying tools when necessary, along with the parts. So after buying the items above I'd save the bike budget up until you've got enough to give it a service.
Once your new bike fund gets to around 1/3 of what you think you'll spend on a new bike, start looking at used bikes. Apart from getting you much more bike for your money, this would get you on your event bike sooner, which is good for training. But it can take time to find one that's right. Do this even if your heart's set on a new bike - to get used to what's available.
Clothing: be comfortable, be seen, keep the wind and rain off - use what you've already got as much as possible. Unlike some others I say hold off the padded shorts etc. until you feel the need, and you may find that they're no benefit on anything other than a road saddle (my hybrid has a 4 hour saddle with or without padding, even 6 at a push, but my tourer needs padding over about 2 hours). Keep your eyes peeled for good deals (again, get to know local cyclists).