When my customer pays for a booking on my website, does the money come into my own account first?

Yes, in most setups the money lands in your own account first. When you take card payments through your own payment gateway on your portal, the customer is paying you, and you then pay the supplier for that booking out of the balance or the credit you hold with them. Your customer sees your own agency name against the payment, not the name of a company they have never heard of.

There are cases where it works the other way round. Some suppliers, and some airlines in particular, insist on collecting the payment themselves at the time of booking. With a deposit arrangement you top the balance up with the supplier in advance and every booking draws down from it, so the money moves in the opposite order. Which of these applies depends on the supplier, on the product and on the terms you agreed with them, so it is worth checking supplier by supplier rather than assuming they all behave the same way.

The timing matters as much as the direction. Card money normally takes some working days to settle from your gateway into your bank account, while the supplier often needs paying at the moment the booking is made, so you need enough working balance to cover that gap. Refunds usually travel back along the same path they came, which means the supplier refund has to reach you before you can hand it back to your customer.

Before you go live, ask each supplier how and when they want to be paid, and ask your payment provider how long settlement takes, so you know how much working money to keep aside.