Bricklaying is an OK DIY job if your are generally OK with DIY. Just take your time. Key is getting everything square and plumb (use a big level and string lines) and getting the mortar mix right. As a beginner you will be pretty slow so just mix a small batch of mortar or it will go off mid job. Also don’t do too many courses at once, fine for the pros but better to go steady on your first go, so it will take a few sessions.
As for tying it in you can knock out a few bricks and tooth it in (proper job, tricky but not that tricky) or use a wall starter kit (very easy)
It’ll be cheap. Sand and cement is cheap. Bricks can often be had cheaply from ebay or facebook/freeads but you’ll need to be able to collect or local builders merchant, but they are a nightmare to get a price from as a DIYer ime. I ended up getting mine from the local merchant but bought through a third party as it was cheaper and moderately hassle free.
Depending the bricks and how many tools you need to buy it’ll cost a few hundred but you could probably do it for well under £200.
Its a very satisfying job so I would, but I don’t think the financial savings justify it