For taking lambs for slaughter, we approach a butcher first, he then says when he will be able to chop them up, so we work backwards to when to book them in for slaughter, allowing a few days to hang them (this tenderises and relaxes the meat). Not all abattoirs will do small batches of lambs, so it could well be worth your while to go along and ask at the office (mornings, as they knock off in the early afternoon, to be back at crack of sparrows the next day). Many abattoirs only kill sheep on certain days, or pigs or whatever. You could also ask to see round the facility and check for yourself that they kill in a humane way that you are satisfied with for your animals.
Don't try to pull the wool over their eyes by making out that you know what you are talking about - they are far more likely to be helpful if they know you are a beginner, so they would use more understandable language and explain things more carefully.
It is well worth your while getting to know the staff at your abattoir and showing them that you care how your animals are treated. The staff are unexpectedly caring people and like the sheep breeders to be likewise.
The butcher pays the bill when he collects the carcases then adds the slaughter charge to your bill.
There are many rules and regulations about taking sheep for slaughter which you really do have to stick to. In particular following your question, if you slaughter your lambs yourself then you are the only person who can consume it. So slaughter charges are something you have to allow for in your accounting. Half an hour to the abattoir is relatively close !!
For starting a flock, think carefully before you dive in. There are two types of sheep rearing - one for purely meat and one for raising breeding stock to sell on to other breeders, and those that don't make the grade go in the freezer.
There are all sorts of breeds of sheep available, so have a look around and find out which ones appeal to you, which ones are most suitable for the type of enterprise you want, which breeds have fewest problems, which would provide a good pelt to get tanned to add a bit extra to your margins and which you can afford. There is quite some controversy about whether to breed from orphan lambs, and some birthing and mothering problems are hereditary, so your lambs may be carrying an unsuitable gene - and if you are picking up orphan lambs from a mart you won't know their history or how they have been cared for before appearing in the mart. I must admit I wouldn't deliberately buy in orphans from a mart ever, but then my enterprise is breeding pedigree breeding stock.
Just looking through the posts on TAS there seem to be a disproportionate number of problems with orphan lambs which have been bought in, with accompanying vet and knacker bills
In the autumn, when lambs have been weaned, there are a number of sales of breeding sheep, where you could pick up a bargain. To make any enterprise profitable it really is worth investing initially in good stock, of whatever breed you choose.
Making a quick profit from a few sheep will never make you rich and may lead you to try and cut corners at the expense of the animals' welfare, but in the long run the more attention you pay to their welfare the better they will perform.