Part-time workers' entitlement to holiday is pro rata – so if you normally work three days a week instead of five, you get 16.8 days a year (i.e. three-fifths of 28 days).
You may need to wait until you have worked enough days to accrue your holiday before you are allowed to take it. For example, if you have only been in the job one month, you may only be allowed to take one twelfth of your annual holiday allowance. To take a longer break, you would need to work longer to save up enough leave.
This entitlement may or may not include public and bank holidays. If you have a rather mean employer, they may decide that bank holidays form part of your 5.6 weeks leave, rather than count them as an extra day off.
Even if you do get public and bank holidays on top of your leave entitlement, there is not necessarily any entitlement to be paid for these days, so you should check your contract to see what you are due.