This area is full of all sorts of creatures, but more specifically a Rogue Tomato, who is near the cliff, which is close to where you start off.
Boss: Rogue Tomato
This isn't really a boss, but you have to kill it to progress the storyline, so kill him off with a few swipes of the sword. Unless you're clinically inept you should not find this too difficult, once you've whittled through a certain percentage of HP the killer tomato will jump off the cliff and start regaining HP. Run down after it and finish it off.
You will get a few Licence Points for killing this one, if you're not sure what to spend it on, then try out some white magicks like Cure, which is very useful, especially at the beginning.
Run around, level up, find and open some of those chests, and once you've had your fun head back to Rabanastre and the Sandsea for your reward. Talk to Kytes, who isn't far from the Save Crystal, to trigger a cutscene.
Head back to the Sandsea and talk to Tomaj to get your reward: 2x Potions, a Teleport Stone and 300 Gil. Tomaj also suggests you have a word with a certain Bangaa in the north end of Rabanastre. If you've been exploring around, it's the Bangaa in the north end that was previously refusing you entry to a certain building. There aren't that many buildings in the north end, so if you're not sure which one it is, just look for a 'Conspicuous' Bangaa outside of it that is now allowing you to enter the building.
While you're still at the Sandsea, you might as well check the message board again. There's a new bill posted for the Thextera, a mutant wolf that's been prowling the Westersand. Speak to Gatsly, who is conveniently located in the Sandsea, to start the hunt.
Speak to the moogle Montblanc to 'join' Clan Centurio, a clan dedicated to the hunting down of monsters as per the instructions of petitioners who are willing to play a nice sum for the service. If you talk to Montblanc again you'll get 3x Potions for joining the clan.
Leave the clan hall and head for Lowtown, which is marked by the staircase symbol on your map - there should be one to the east of the Clan Hall.
You can explore for the various treasure chests and shops around here, although if you want to obtain a brilliantly overpowered weapon later in the game you must not open the chest that's across the street from Old Dalan's place (see the Zodiac Spear for the secrets and sidequests section of this guide) - the contents are usually inconsequential anyway.
Speak with Old Dalan, and you find that your next destination is the Giza Plains, and it's recommended that you get a map from a Cartographer's Guild Moogle, one of whom is located in the north-western corner of the Southern Plaza.