No public transport in the town, but given that it is fairly small there is no real need for it. You could just flag down any car on the street, most would be happy to bring you around the town for 2-5 somoni.
In town there is nothing to do. Internet is available for 10 somoni per hour at ACTED during weekdays. The surrounding are really beautiful, and several trips can be made from Murgab. Most require your own transport or many dollars.
For those keen on walking, you can walk to Gumbezkul in the Pschart valley (35 km), sleep there in a yurt (25 somoni), hike up the pass (4731 meter) and walk all the way back to Murgab (31 km)
There is a mobile network in Murghab available. For internet, you can buy Beeline's sim 'internet' (on others promotion internet is too expensive!). Then apply for monthly 3GB package just for 6$. (Recharge your account, then issue USSD-code "*110*820#"), wait until 1:00 am for activation. It's better to have Indigo sim also, since some areas in Pamirs are better serviced by Indigo. Sim cards should cost 1-10 somoni, depending on how much money it already charged.
There are only basic food supplies in Murghab. You can buy rice, flour, oil, alcohol, sugar, potatoes, onions, some dried fruits, simple candys and cakes in bazar. But not much more. Few basic restaurants around bazar are open only in daytime. Most hotels and homestays could provide meal, often included within the price (1-5$ per time per person).
There are both homestays / guesthouses and hotels. Prices can be bargained for, depending on the inclusion of food and time of year. Expect to pay somewhere between 4 and 20 dollars.
Sukhrob Guest House. Friendly and hospitable familly (english spoken) where you have the option between dorms in tajik style (mat), kyrgyz style (yurt) or ordinary style (beds), or a family room. The place is not far from the bazaar (Samoni street), on the other side of the main road (uphill). Just ask around for Sukhrob!
Homestay of Talay. £12 per person per night incl dinner and breakfast.
Local hotels exist, one is not so far from bazar (ask for 'gostinitsa') In May, 2012 they asked for 10-15$ but agreed for 32 somoni (7$) without food, shower, or room heating. Either of these costs few dollars extra.
There are shared vans/jeeps to Khorog almost everyday from the bazaar, 100-120 somoni per seat (May 2012). They wait for passengers and depart when full. Weekends there are fewer passengers. Small chinese-made vans should be cheaper, since they use less fuel, than normal jeeps. If you want go to Dzhelondy, Khargush, etc... they would ask the full fare to Khorog, so better option is hitchhiking. Just head south along the highway, cross the bridge, walk up to the checkpost (3-4 km from the center). You can hitch the trucks coming from China or some vans, passing by. 40-50 somoni to Dzhelondy or 60-80 to Khorog should be enough. From Murghab to Khorog there are more than 10 chinese trucks on average daily (if the border is open), and few other cars. On the Khargush-Langar road there could be no traffic for days. Dzhelondy really worth a stop for few hours to soak in a nice hot springs, it's so pleasant after cold and windy Eastern Pamir. There are few hot springs in Dzhelondy area, most can also provide food and basic accommodation, Then its not hard to find onward transport in Khorog direction
Shared jeeps to Osh are less frequent, departing only when they get enough passengers. Probably 1-2 every day or every second day. Rate is 150 somoni to Osh. You should come to transport stand ('stoyanka') at the bazaar at 7-8 am, as they start early sometimes. Or even contact driver from the evening. But don't just rely on one driver, who 'promise' you he will find passengers tomorrow, probably he won't and so won't go. You better wait at the transport stand, sometimes there will be some occasional transport. Hitch hiking is also possible, there are some cars which start or terminate at small settlements between Murghab and Kyzyl-Art. Each 40-50 km at most there are settlements or road worker's houses along the highway, where you can sleep or have a tea. From tajik border post you can walk to kyrgyz border post in 3-4 hours, all the way down, nice walk. And from kyrgyz border post is a bit longer hike over 25km, across Alai valley to Sary Tash, where there is a lot of transport to Osh.
If you are hiring a car all the way to Osh, make sure its either a Kyrgyzstan car, or driver has all the necessary papers to cross the border.