What sort of insulation should I use in my old house?

If you have an old house then you need to take care with the type of insulation you use.
If your house was built before 1930 it is definitely breathable. If built between 1930 and 1940 then it may be breathable. You can tell by the build - if it is single brick (no cavity wall) and has a suspended floor then it is breathable.
Breathability is all about the ability of vapour to move through the structure. If you block that movement at any point you can end up with interstitial condensation which will damage the brick.
So in a breathable house you need to build, extend, retrofit in a breathable way. That means using breathable / vapour open insulation materials.
So don't let the builder tell you you can use PIR. It is petrochemical based and definitely not breathable, so you will end up with problems.
There are plenty to choose from as this video will show.
You can explore the options further on the website of Unity Lime: unitylime.co.uk
Any questions just let me know.