The problem with kits is quite often you end up with a lot of censored you never use, moreso if you already have alen keys and spanners and mole grips... So it's important to chose carefully.
Assuming you already have
Then there's not much day to day stuff you need, but things I've bought that have been useful are
Bike multi tool (to carry around)
Chain link extractor tool
A KMC missing link or two
Some bike specific chain lube
Bike cleaning brushes, sounds daft but they are dirt cheap on amazon and handy.
A track pump for home use.. I just use a really cheap one but it makes pumping tyres so much nicer than a mini pump
A mini pump for your frame
If you get more involved down the line then
Crank extractor wrench
Suitable grease for bearings.
Next thing on my list when I get round to it is a digital tyre pressure guage.