You could easily spend a fortune in a bike shop and get lots of fancy accessories, but not everything is essential. Over the years these are the items that I have found to be the most useful.

Panniers
These are bicycle specific bags that clip on to racks at the rear or front end of the bike. They are essential for overnight trips so that you can carry your belongings. This photo shows my bike kitted out with rear panniers fitted onto a rack on the back of the bike. 

My bike with panniers

I have Altura panniers and they have lasted me for about 15 years, so the initial outlay will be paid back. My panniers have waterproof covers that are 100% effective at keeping everything dry. This is the most important feature when selecting panniers. You need them to be waterproof. Expect to pay around £60 for a pair of good quality panniers.

Lubricant for the chain

I always carry a bottle of lubricant for the chain. When your gear changes are no longer smooth and the chain sounds like it is not running as smoothly this is a sign that you need to add lubricant.  Regularly adding lubricant to your chain will make your chain last longer and reduce the chances of it needing to be replaced as often. You can pick up a bottle of lubricant for less than £10 and it should last you a few years. I currently use Muc-Off wet lube.