Palmyra is in Syria.
Palmyra (the Roman name) is known as Tadmor to the Syrians. Both mean the same thing - date palm. The name comes from the lush oasis adjacent to the city which is home to some million date palms. It is the only oasis in Syria and perhaps the only truly tourist town.
Palmyra sits on the standard tourist trek around Syria and should be considered in this light, being one of the most stunning archaeological sites in the Middle-East, if not in the world. Intense competition for business amongst local outfits can make the experience somewhat overwhelming to the traveller who has come from the North, and has enjoyed a relatively 'quiet' trip thus far. The major tourist attraction of the area is the stunning ruins - the most famous and well-preserved of which are the Temple of Bel, the colonnade, the funerary towers, the hypogeum of 3 brothers, and the Arab castle, all in this very particular desert context. All are within a few kilometers of each other.
There are no ATMs (that accept international cards - there is one for Syrian cards) in Palmyra or even a full-service bank. Hotel Bel (on the main street) will do advances on both VISA and Mastercard for a 20% commission. There is a local exchange office by the museum which will change foreign currency but will not change traveller's cheques. Make sure you have sufficient cash, Syrian Pounds, US Dollars, or Euros, for your time in Palmyra.
Note that, as per usual, the Syrian Commercial Bank offers terrible rates and adds commission. You'll get a better deal by checking the rates online than changing with the shop owners in the Souq.
Souvenir shops abound on the main street with all kinds of jewelery, handicrafts and other wares typical to the Bedouin places, many of which may be nice to buy, depending on what you like. Shopkeepers (in a town with few sources of income other than tourism) are masters in the art of making you part with as much money as possible, so it would be wise to only carry as much money with you as you are willing to spend on souvenirs (and food - see below) to avoid being talked into spending all or most of the money you have for the rest of your trip in Syria. Typically souvenir shopping would happen around dinner, as many of the restaurants and shops are in the same main street and you would happen to walk past them on your way to or from the hotels and restaurants.
On the same street are several stands selling roast chicken (half chicken for take away is 100 SP, hummous 25 SP, salad 25 SP), although you'll be lucky to get those prices.
If you venture on any of the main roads running north you will find felafel stands and other small restaurants selling the typical range of Syrian fast food bakeries selling sweet treats and plenty of convenience stores with drinks and snacks.
The only bars in Palmyra are inside the hotels, such as the Cave Bar in the basement of the Ishtar Hotel. The bar carries good selection of local beers and wines, and you can have your drink in the terrace of the hotel if you wish.
They are all closed and some have been destroyed since 2011 and 2012.
On the main tourist drag, the Hani Internet Cafe inside the Traditional Palmyra Restaurant is conveniently located but charges a pricey 50 SP for a half hour. This may be negotiable in low season. Locals can direct you to an internet cafe slightly north of the centre which only charges 20 SP an hour but has irregular hours.
Palmyra has recently been recaptured by Syrian government forces from Islamic State. Whilst the damage to the site is less than feared, and may be at least partially restored at some point, the main temples, arches and funerary towers are for the moment just rubble. Although ISIS have been removed for now, Palmyra is still a war zone with many land mines and booby traps left behind, and it is therefore strongly advised not to attempt to visit what is left of it at any time in the foreseeable future.
Buses depart frequently for Damascus, Homs, and Deir-az-Zur.The bus station is a little under a kilometre away from the main street, so do not pay anymore than 50 Syrian Pounds for a taxi with your luggage. For other destinations, you may need a private car.
If hiring a private car, you might want to consider side trips to Qasr al-Heir ash-Sharki - a partially excavated Ummayad palace quite literally in the middle of nowhere - and Rasafa, originally a Roman city with heavy Byzantine influence, also used by the Ummayads before being destroyed in the Abbasid era. Rasafa is also of interest for the stone it's built out of, more a quartz-like crystal instead of the usual granite or sandstone. Makes for a unique appearance. This route takes you quite close to the Euphrates, and you can be dropped off in Raqqa, Aleppo, or Hama. Car hire can be price, and the driver still has to get back to Palmyra. Private tours are the real money maker in the Syrian tourism industry, so expect to pay as much as S5000 (US$100), if you're heading for Aleppo or Hama. A bit less to Raqqa.