A lot of good advice so far.
I'd say the first thing you need is a saddle bag of some sort - I like the Topeak Aero Wedge (
here) as you can quickly remove it from the saddle, which I do if leaving it outside the supermarket in questionable areas or at night.
Then you need at least one inner tube, and a good pump - something you can attach to your frame on the bottle cage mounts is good - I tend to use Lezyne pumps but the
Halfords Advanced pump is also really good - I have one for Mrs C. Just leave it on the bike at all times - then you have it when you need it.
The saddle bag should have at least one inner tube (depending on the size of your bag), some instant patches (such as Park Tool, or Skabs) which let you quickly repair a puncture at the road side. Tyre levers (Park Tool, Pedros or the Bontrager Red) - getting
good levers is really important, bad levers make your life harder - you should be able to remove and put on a tyre without them, but it's quicker and easier on your hands to use levers. Have more than one, and know how to use them - tips
here.
A good multi-tool is also important, this largely comes down to preference, I use the
Topeak Hexus X but any good tool here is worthwhile.
Don't skimp on the quality, it's tempting to get cheaper tools, especially when starting out, but if you're commuting and you have a problem then you need to be off quickly.
A good lock is an investment, I've had some of mine for more than a decade - I tend to use a d-lock, the
Kryptonite Evolution Mini 5 is always in my backpack - it's reasonably light for d-lock and I've not had any problems locking my bike to most things. I also carry a separate cable lock just in case. Remember, you're just foiling opportunistic thieves and for those who are equipped to remove locks you just want a better lock than the bike next to yours.
Also worth getting are some decent gloves and a light rain jacket for now - then re-think clothing later in the year when it starts to get colder.