Pengerang is an area at the southeastern tip of Johor, to the south of Desaru. The name is usually used to cover the dock and immigration post at Tanjung Pengelih, the village of Kampong Pengelih and the nearby town of Sungai Rengit.
Sungai Rengit is around 35 km south of Desaru, or 120 km from Johor Bahru. The distance from JB will decrease considerably once the Johor-Desaru Expressway opens. [Update] E22 is now open and the journey from JB to Desaru is now less than 1 hour.
By bum boat
Bum boats operate between Changi Ferry Terminal and Pengerang daily. There is no fixed schedule- the bumboat operators will only leave when there are 12 passengers. The earliest ferry is around 7 am, and the last bum boat is around 3-4pm Singapore on weekends. The schedule varies on weekdays. If you want to leave before 12 passengers have arrived, you can pay the boat operator for the missing seats. The price per passenger is S$11 each way, . Additional $2 for a bike.
The terminal in Singapore is located in the basement of the Changi Point Ferry Terminal. You will need to give your passport to the people operating the desk underneath a large "Pengerang" sign. They'll note your information on a sheet listing the passengers. When this list is full and a bumboat operator has arrived, he will return your passport to (at this point, follow him through security). The ferry from Pengerang to Singapore has the same procedure.
The Changi Village bus terminal is 100m from the ferry. Taking a bus here is the best way to arrive via public transit.
Taxis shuttle between the Pengerang dock and Sungai Rengit for RM 5/person (shared, four person minimum) or RM 20/car (charter).
You can rent bicycles from Tristan Park Group for S$10 per day. If you walk about 1 km eastward on the road leaving the ferry terminal (past the naval base, but before Pengerang), you'll see a sign for Tristan Park. There is a bicycle shop in Sungai Rengit (about 30m north of the Tai Hoe Hotel) that rents decent quality bikes for around RM 15 per day.
Alternatively, bicycles can be rented in Changi Village and brought over on the ferry.
Pengerang was the site of a major WW2-era British coastal battery. The part of it at Pengelih, right next to the immigration point, is open to the public but in a state of nature, with no trails, signs etc. Most of the rest is now used by the Malaysian military. 
There is an ostrich farm just outside Sungai Rengit. The cost of entry is RM15 per person, and includes a brief presentation by the farm's owner (in English). There are several hundred ostriches in various stages of life. The more tame ostriches roam the farm freely, and pictures are encouraged. Outside the farm, ostrich satay is on sale for RM 2 per stick. Ostrich steaks, burgers, and omelets are also for sale. The farm is open until 6 pm daily, and is about 1km east of Sungai Rengit (you'll see a large sign on the highway).
Cycling on the highways is safe, as the roads are well-paved and not heavily trafficked. Most cars keep to about 60km/h. The ride to Desarue is quite flat and pleasant.
Chinese pastries are a popular souvenir.
Road-side stands are commonplace, selling home-made sweets (3 for RM 1), sugar cane juice, and cold drinks. Stalls can be found near the coast in Sengai Rengit.
Seafood! It's the main reason to come here, with treats like lobster and crab much cheaper than Singapore or even JB.
There aren't really any bars in Sungai Rengit, but large bottle beers can be found in many restaurants, including Good Luck Restaurant and Jade Garden.
There are a few basic lodging places in town. If you want more luxury, head up the coast to Desaru's beach resorts.
There's a small police station in Sungai Rengit. The area is quite safe- many people leave their bicycles out all night with a worry of theft. On the roads, drivers are quite courteous and cycling is very safe. Be wary of taxi drivers passing many cars at a time, or speeding much faster than others on the road.