I'm assuming that you will be arriving about midnight when the U.S. carriers arrive. If it were me, I would book a room at the transit hotel as there are no night ferries to Batam. Then in the morning I would take either a taxi or MRT to Harbourfront depending on your budget. Ferries leave fairly frequently during the day, although on Fridays and weekends, they fill up pretty fast. Take any company. There are multiple ports on Batam, so it is best to know in advance which where on the island you desire to go. Taxiis are now coupon controlled, so no more bargaining. Taxiis to Nagoya in January were 45000 rupiah. In January RT SIN-Batam was about $27 U.S. Visa on arrival is $10 for approximately 1 week, $25 for a 30 day one. Visa are payable in U.S. dollars and dollars are quoted above. You can also pay with Sing $$, but the Visa is a couple of U.S. dollars higher that way. Be sure to get your visa before getting in the immigration line.

If you decide to go to Bintan, then take a taxi to Taheh Maruh terminal where the ferries dock. Buy your ticket and away you go.

Note: There is a one hour time difference between Singapore and Indonesia. Travel time to Batam is approx. 1 hour. Bintan is approx. 1.5 hours. The last ferry back is somewhere around 8 p.m. The ferry companies have web-sites. One is Penguin, another is Batam Fast Ferries.