I would buy a Haynes manual for your car. If you can't get one from Haynes or Amazon, eBay should have a second hand one for very little. This will give you all the advice and guidance you need on maintaining your car. It will also provide some help on buying tools. It should also allow you to do a lot of 'inspection' work yourself - so I wouldn't bother taking it to the professionals for that unless you find something wrong that you don't think you can handle yourself.
The light thing is normal. As long as they go off when you start the car, that's fine.
In terms of tools, I'd start modestly and then buy extra stuff when you need it. But these are things you will need to get started:
Socket set: I would get one that only has metric sockets (as you won't have much use for the imperial ones on modern cars). You will probably need at least the following size sockets: 8mm, 10mm, 12mm, 13mm, 14mm, 15mm, 17mm, 19mm, 21mm. German cars use a lot of 16mm but they are fairly uncommon elsewhere. You may need to fill the 'gaps' in as you do more work. Try to get one that is aimed at mechanics as this will have useful extras that you will need such as 'deep' sockets and spark plug sockets. My best advice would be to get a fairly large one with 1/2", 3/8" and 1/4" ratchet wrenches (the wrenches are still measured in imperial - the sockets themselves will still be metric). If you can't afford this, a joint 1/2" and 3/8" would do, or maybe just a 3/8" set at a pinch. You don't need really expensive ones but avoid very cheap sets. Expect to pay at least $50+ for a decent set. This one is aimed at mechanics and includes some of the other stuff in the list below. It's at the cheap end of the market but should be fine to start with for occasional work. But if you can afford a better one I would spend a bit more.
Screwdrivers: flat head and Phillips (cross-head)
A set of hex keys ('Allen wrenches') like this one
A set of 'combination' wrenches with one 'open' and one 'closed' end like this.
You should also get at least one torque wrench. This makes sure you tighten nuts, etc. to the correct tightness. I may seem like a luxury but I assure you it's not. You'll probably need to cover a range of torques from about 25Nm to 120Nm for basic tasks.
Other wrenches such as a water pump wrench, a 'mole grip' and some pliers would also be useful.
A rubber mallet is also useful.
You will need a jack and axle (axel) stands, too.
Good 'medium range' brands like Draper or Laser will be fine and probably last a lifetime for occasional use.
That should set you up. I'm sure I've forgotten a few things and other people will add useful suggestions. It may seem quite a big outlay on tools at the start but you'll soon get your money back with saved garage bills. You'll probably become more attentive to your car's needs too and prevent many major problems arising. It's not very hard to do basic oil changes, filter changes, brake work, etc. and you might even find that (like me) you come to really enjoy it.
Good luck!