The whole peninsula has plenty of beaches even if they don't belong to the longest in Scania. Båstad itself has a beach as does Torekov and the southern coast of Bjäre is full of a mix of beaches and nature reserves.
As with the rest of Scania, this is an equestrian landscape and you can go riding (with or without included picnic) in the countryside from several villages. In the hills above Kattvik, an Irishman has his trekking centre which is one of the most famous but there are also Icelandic horses. Ask the local tourist office for an up to date list of stables.
Equipment: You can usually borrow hats, just bring good footwear and comfy trousers.
Swedish Open takes place every July when Båstad is suddenly filled with international tennis stars and their fans. Book your accommodation well in advance if you intend to visit during this time and the weeks around it. There are also other tennis tournaments here such as the Kalle Anka (Donald Duck) Cup for promising youth players. Almost all famous Swedish tennis stars have won it :)
For a week in July every year you can see the tennis elite fight for first place in the Swedish Open.
Equipment: You can never really rely to much on the Swedish weather. Some years it's clear blue skies, others it rains alot. Games are re -scheduled when it's too bad.... a rainproof jacket or umbrella is a good backup unless it's bright sunshine and 30 C.