Hi there,
What you need is the i-link travel card. As you've found, the Translink website isn't particularly obvious or helpful to use so here's the link you need:
https://www.translink.co.uk/usingourservicesandproducts/ticketsandtravelcards/ilink
It covers all of the Translink trains, the Enterprise train within NI, all local buses within NI and the Goldline coaches. (It doesn't include the private tour buses around Belfast though.)
It costs £16.50 per day or £60 per week.
There are no peak time restrictions and there's no senior related discounts as if you're over 60 and a NI resident it's all free.
You can buy the card in advance and have it delivered to you and it doesn't start until you first use it. You can buy one in the Tourist Info centre in Belfast but I'd recommend getting it ahead of time.
Have a good trip over here!
Mr Toad