You can do most of what you need without much/any special kit. Local playgrounds, benches and hand rails can be useful.
There are various types of:
Chinups,
pullups,
pressups,
dips (using a bench or between 2 chair backs)
bodyweight squats (2 leg or 1 leg)
abdominal exercises
etc, etc..
As others have said, strengthening the core/trunk is the most beneficial thing though
-everything hangs off it.
The above exercises will also work the core muscles if done properly.
For extra weight, sandbags are fantastic. 50Kg of sand costs about 6 quid from B&Q, and much more beneficial/awkward to pick up and move around than a 50Kg barbell.
The internet is full of ideas.
The main thing is to do the exercises regularly and stick at it.
I like to mix it up a bit.
Pilates is ok, but most classes are not hard, according to my wife
-I suspect that most people don’t want their exercise classes to be too difficult.