You can access some of the deals from the UK by getting on the Thomson holidays site and a few others
We booked a few days in the Algarve last year (we did not need air as we were on an extended vaction through Spain and Portugal.
The Thomson site will provide you with details on holiday packages, air, as well as let you book individual hotel stays.
When we first tried to book, the site would not accept our booking because we did not have a European address. We changed our address to that of the hotel where we were staying in Seville prior to going to the Algarve. Once we did this, the 'system' accepted our reservation, charged our credit card, and spewed out a reservation without any issues. We also found on our arrival at the hotel that the rate included 'half board'. This turned out to be a complete buffet breakfast and an excellent evening meal. I am not certain what would happen with a reservation that included air as the information required for air is much greater than that required for a simple hotel reservation. We have also booked on line tickets for discount carriers such as easyJet. No issue whatsoever and far cheaper than the full service carriers. You just have to be aware of the luggage weight restrictions. We are light packers so this is not an issue for us.