Do you use any invoicing services? I've heard of services like FreeAgent and Crunch but am not sure what's best. I would prefer to not hire an accountant but I'm afraid, maths isn't my strength so would like to go for something easy to use and not too expensive. Any suggestions?
No, I do it all myself, keeping track of outstanding invoices, incoming payments, and expenses on a plain old spreadsheet. Some time after the tax year ends, I go on the HMRC website and start a self assessment. The website does a good job of walking you through the process and handling the math for you, so as long as you know how much you made and how much you spent on expenses it's pretty easy.
The thing to remember is even though the tax year runs from April to April, you don't have to file until January 31st, so you have a loooooong time to get it all together and get it done.
Also, I know the UK tax year runs from April to April. I won't be filing my US taxes this year because I didn't make a lot of money last year as I only worked for two months. Do I need to file now that I've registered as self employed (even if I don't think I'll make any freelance income before the end of the UK tax year?
I wouldn't, especially if you didn't/won't make any money before April anyway. My guess is if anyone ever asked you could just explain you registered in March but didn't make any money until you started working in April (or whenever).
Again, I'm not a tax person, so I may have just advised you to break the law, in which case I plead ignorance and throw myself on the mercy of the court.