It's good to hear you are planning a trip to Ireland next month.
Staying in Bray, you will be able to use the DART to travel north to Dublin. The Leap Visitor Card
comes in handy here. You can choose from a number of different options, including a 3-day visitor card which costs €16. There is also a 1-day card and a 7-day card.
The DART doesn't travel as far as Arklow, however. There is a number of public transport options here, including by train
In terms of travelling between Arklow and Avoca, you could avail of the Bus Eireann bus service
I hope this information is helpful to you.