If I buy an inflatable kayak, can I just go up and down the river ?

[PotteringAboutIn]

as long as I can get in somewhere

Depends who owns the river. And what country you are in.

Most rivers in England require a Waterways license...but its rare to be asked for it.

Yes, but best to go upstream first, and then come back with the flow.

Oops sorry, I didn't know you needed a license.

Just after the war, my dad and his friend ( aged 12/13) bought an army surplus inflatable dinghy, took it in a rucksack on the bus to a place upstream on the river Trent and floated/paddled it back downstream again.
I'm glad they never fell in as the water quality was terrible. Also surprised my dad lived long enough for me to be conceived!

To answer your question. Yes you probably can but please educate yourself about currents and plan how/where you will get in and out of the water. Rivers can have much stronger currents than it appears from looking at them!

I thought this was going to be a metaphor for some relationship difficulty...

Look at Paddle UK, explains it all there.

https://paddleuk.org.uk/waterways-licences/

Wasn't certain if small paddle craft need a license but checked and there is the result.

I remember when I was young, we had a boat on the river and the licence was checked as we went through locks.

I should get one with a puncture repair kit, and take a pump with you. To avoid the whole thing being a deflating let down.

You need a waterways licence. I can remember sticking them on our Scout troop's canoes and, on the River Wey at least, they were checked at locks.

I’d get some lessons first. You need know how to paddle effectively, what to do if you capsize, etc. Also ensure you have a buoyancy aid. Enjoy it but don’t become another news headline!

Beware rapids, weirs, fishing hooks/lines/nets, culverts and leptospirosis.

They are clamping down on unlicensed river users in the south-west. People have been fined recently and they are fairly hefty fines!

If you join paddleuk I think the license is included in the membership.