The Visa on Arrival that was introduced in 2010 and never made available to the US Passport holders anyway, was phased out and replaced by E-Tourist Visa as Printman says, and also extended to US Passport holders. This is similar to what Australia offers to US Passport holders too. Here are some details from the proverbial horse's mouth:
https://indianvisaonline.gov.in/visa/tvoa.html
BTW, while I am an Indian native, I am a US Passport holder and an Overseas Citizen of India. The normal visa regime does not apply to me. By virtue of being an OCI I have a Lifelong Multiple Entry Visa and once I am in India none of the police registration and other rigmarole for staying for extended periods does not apply to me, nor do restrictions regarding property ownership etc. (except for certain specific exceptions). But in order to exercise these lesser restrictive rights equivalent to those of regular Citizens of India, I have to carry my OCI Id card around with me, and use it for entering India rather than a y other visa regime.
One thing to note though is that this facility is available at only 9 airports of entry. If you plan to cross the border by land anywhere, like say to go to Nepal by road, or Bangladesh by road or rail, you need to apply for and get a regular paper, possibly multiple entry, tourist visa.
To bring it back to trains, in particular you will need this (Indian paper visa) if you plan to travel by the international Maitree Express between Kolkata and Dhaka or vice versa. Bangladesh has a Visa on Arrival facility which is allegedly available at all ports of entry, however people who have tried to use it at Darshana (across the border from Gede) have had mixed results while attempting to enter, and no problems while exiting from a stay in Bangladesh after usin VOA entry elsewhere.
There is a proposal being worked on to move the immigration processing for Maitree Express passengers to Kolkata and Dhaka Cantonment, and running the train as a sealed train from Kolkata to Dhaka. This is also in conjunction with converting the train to a fully air-conditioned service, but it is a few years away and possibly will also happen with it becoming a daily train. When that happen the situation with using Indian E-Visa and Bangladesh VOA more reliably will also likely come with it. In addition there are two other proposals in the works:
1. Reintroduction of the Barisal Express from Kolkata to Khulna via the Benapole - Petrapole border crossing near Bongaon Jn., mentioned above. This train was discontinued in 1965.
2. A through train from Kolkata to Agartala in the Indian state of Tripura, across Bangladesh, via Dhaka, possibly to be run as a sealed train, reducing the journey distance from some 1600km through India today, to some 600km. or so. A non-sealed bus service on this route has just been introduced. The crossing into Bangladesh would be at Gede. The crossing back to India would be on a brand new Broad Gauge link that is currently under construction between Agartala in Indian and Akahura in Bangladesh. This incidentally is part of the Southern Trans-Asia rail link which someday is supposed to connect Turkey to Viet Nam and Southern China through the Indian subcontinent.
In addition, there are rumors floating around that some preliminary discussions for reinstrituting a train on the route of the pre-independence Darjeeling Mail between Kolkata and New Jalpaiguri has taken place. Such a train would cross into Bangladesh at Gede, like the Maitree Express, but would proceed north from Parbatipur to cross back into India at Haldibari.
One other possibility is that the handover of locomotive power moves from Darshana to Gede, which would allow Indian Railways to run these trains under electric power to Gede and handing it over to Bangladesh diesel power there. Kolakat to Gede is electrified (25kV 50Hz), and the electrification ends just short of the Bangladesh border. But this change is not that critical. It is interesting to speculate that had there been no partition it is more than likely that both Darshana and Jessore would have been electrified and would have been the end points of Kolkata Suburban service, which today terminates at Gede and Bongaon respectively..
The situation with using the Samjhauta Express from Delhi Jn. to Lahore, to cross the land border from India to Pakistan may be somewhat more fraught specially for US citizens, for reasons other than mere border formalities. Afterall Pakistan has always been a much closer friend of the US than India
, and such intimacy comes with its own issues.
Anyway you can read the country specific warning issued by the US Department of State:
http://travel.state.gov/content/passports/english/alertswarnings/pakistan-travel-warning.html
