Fixed rate accounts are usually opened with a lump sum that can't be added to. If you're only just starting to save monthly, do you have a large lump sum to lock away yet?
If not, look at regular saver accounts to start with. They have a maximum monthly amount you can deposit (often up to £200 although it's £500 with Coventry Building Society). They'll have a set duration, eg 1 or 2 years. Check whether you can withdraw money if you need to. Shop around (Coventry is currently paying 2.4%) and check your bank too - some of them have a better account if you have a current account with them.
How much of a savings buffer do you have already? Might you need to access some of the new savings in an emergency? Fixed rate accounts can't be touched at all until maturity (apart from if the account holder dies). ISAs are worth considering as they have to allow you access (albeit with a 'penalty' of loss of interest equivalent to a certain duration). At the moment interest on ISA fixed rates are slightly lower than non-ISA fixed rate accounts, but you might have to pay tax on the interest on non-ISA accounts, although you'd have to have a lot of savings for that to kick in.
Some banks/building societies with consistently decent interest rates:
Coventry Building Society
Leeds Building Society
Kent Reliance
Re Pinksunsets question: no, there's no limit as to how many fixed rate accounts you can open in a year (as long as they're not ISAs). But you might be limited to one of any specific account with an individual institution, eg you can open a 1 year fixed rate and a 2 year fixed rate account but they might not let you open more than one of each kind. Well, not during each issue - fixed rate accounts are available for a certain time and then closed to new applicants. And then a new one is launched with a different interest rate. The building societies often number them, so the current 1-year fixed rate might be Issue 17, and the next one will be Issue 18. You might be restricted to opening one account in Issue 17, and when that closes and Issue 18 is available, you'd be able to open another one.