Should I use slime-filled inner tubes or anti-puncture lining?
If you have a lot of thorns around, slime is probably not going to save you. Perhaps tubeless tyres mounted in a tubeless setup could, but they are more complicated and potentially incompatible with your old rims. Tubeless also uses something like slime (most often a latex-based sealant) but it all works much better, because there is more solid rubber in a tyre so it is much easier to seal the hole in a tyre rather than in an inner tube.
Because you have a mountain bike, it might be relatively easy to mount a tubeless tyre even on a non-tubeless specific rim. The wider the rims and tyres are, the easier it is (not talking about fat bikes). However, it is a large step you might not be willing to undertake and it does require some regular maintenance.
A tyre that comes with a good solid anti-puncture lining will be much more certain than slime if you encounter many thorns.