I'm 6 months pregnant via donor insemination at the moment. It was really overwhelming trying to get useful information so I hope what I found helps you!
Your options are:
Known donor at home
You can use websites (PrideAngel comes to mind but only because they spam every gay women's website in the entire UK, constantly) which is like a dating agency for donors / donees. Or find a friend. There are some genuine men who just want to donate, often because they have a friend or family member going through this and want to help.
Pros: It's cheaper and you can, by arrangement, get them to donate on consecutive days, improving your chances of conception (e.g. inseminate 4 days running rather than just once.)
Cons: As well as the genuine men there are some total weirdos who will try to persuade you that the "natural" way (i.e. sex) is more likely to get you pregnant. One was recently convicted of sexual assault after the donee discovered he'd actually had a vasectomy and was just using this as an excuse for a shag. You would have to arrange any STD testing or hereditary illness testing yourself. You have no idea how motile their sperm is.
Caution: Legally, in the UK if you don't use a clinic, then the donor is the father and will be entitled to contact and even residence of the child (unless you are in a civil partnership with another woman). Having a paper agreement is next to worthless. You would also be entitled to go after him for maintenance payments. The man who seemed so happy to be "Uncle X" may change his mind if he disagrees with how you're bringing the child up. These sorts of agreements can go horribly wrong and end up in tears and litigation. I wouldn't use a friend, no matter how trusted, because of this.
Known donor via clinic
Your only option with a UK clinic as all UK sperm donors are known donors. You browse a catalogue of men (yeah, weird) and decide whose sperm you want. Then get the clinic to store the sperm and inseminate you when you're ovulating.
Pros: The sperm is trusted, washed, checked for nasties. The insemination is done straight into the womb, so none is wasted. You don't have to make eye contact with a man holding a jam jar of jizz.
Cons: I hope you are very very rich. UK clinics are expensive.
Unknown donor via clinic
Only available outside the UK. There's about equal research shows that donor conceived children do better with a known donor and with an unknown donor so I haven't put that as a pro or a con.
Pros: Same as with a known donor, plus unknown donor sperm is cheaper than known donor.
Cons: You have to travel.
What I did - I went to Vitanova in Copenhagen for IUI. My personal view (and it is personal) was that the UK clinic I saw was running a business whereas Vitanova was running a service. The UK one wanted me to have my tubes flushed with blue dye etc as though I was there for fertility treatment, and really all I needed was the sperm! But I know a lot of people are reassured rather than put off by the UK's more medicalised approach.
With a known donor you would choose your sperm and tell them at the initial appointment (can be done by phone) who you want, and with an unknown donor you just give them race / height / hair / eye colour requirements. You need to have an STD test and a certificate confirming you're immune to rubella and a couple of other things. Then you just wait to ovulate and when you see that smiley face on the testing kit, hop over to Copenhagen (about £100 term time but can go up to £300 in holiday time). You can get there, get inseminated, have a look round the museum and be back on the plane the same day.
If you want an unknown donor or a known donor but less expensive, and don't mind the last-minute travel, then I can highly recommend Vitanova, they were very welcoming and reassuring - and they speak perfect English!