The best company to test with - assuming you are in the UK - is Ancestry. This is because Ancestry has the biggest database of UK test takers, so you are most likely to get a match.
Ancestry will send you a test kit with a tube to spit into, fill the tube, send it off. Wait about 6-8 weeks. When the test has processed you will then get an email telling you that your results are ready.
And this is where it gets complicated. Ancestry will present your matches with the closest first, down to the furthest away. The "closeness" is measured in Centimorgans, which is just a number. This chart will help explain the numbers you are seeing and what they might mean.
dnapainter.com/tools/sharedcmv4
So best case scenario you get a match around 1800 centimorgans and Ancestry says it's a half-sibling, or an aunt/uncle. You can then CAUTIOUSLY contact that match and say something like "Hey, you're a close match to me, want to try to figure out how we're related?" And they might reply, or might not - lots of people get the kits for Christmas, do it for fun and completely lose interest.
A more normal case scenario is that you don't get close matches at all, and you have a few matches in the 200ish centimorgan range which could be second cousins, half first cousins etc and in this case scenario traditional genealogy is required to start putting together family trees and understanding how the match's family is structured and where you might fit in. In your case this is doubly complex as you don't have that much information about your birth mother OR your birth father and your first job is working out whether any match is on the paternal or maternal line.
So in summary:
proceed with caution and be patient waiting for results
contact matches with a very neutral message to start with
be prepared for them not to contact you in return
get help from a DNA facebook group or professional genealogist if you're confused.