Walk south-east (right if you're looking towards the sea) and you'll head towards Las Marañuelas beach and the harbour. Walk north-west (left) and you'll find the La Lajila natural pool and the coastal pathway that runs through Patalavaca all the way to Anfi beach.
Once you get to the harbour entrance wander in for a look at the traditional fishing boats, or carry on walking left; the road takes you round the corner to where the huge Tuesday market is held. That great big grey thing ahead of you is the cement factory. It'll be gone one of these days.
For a first meal, pick any of the restaurants close to the beachfront from local dishes and seafood. Taste Mesón is a superb option with great views of the beach and harbour. For chap and cheerful local seafood, try the fisherman's guild restaurant or Cofradia within the harbour.
One of the best things about Arguineguín is that it's the only large town in the area and still has lots of local shops and bars. Wander the streets between the main road (the GC 500) and the sea and you'll find everything from shoe shops to printers alongside cafes and restaurants.
Inland from the GC 500 Arguineguin is largely residential with lots of villas and complexes owned by Scandinavians. The main shopping centre, with everything from a curry house to a Scandie rock bar, is just next to the Whale Tail roundabout.