There is one popular border which 90% off overlander and backpackers use to Cross from India to Nepal , Called Belaniya (firts town is Bhairawa in Nepal).
its basically no mans land and you can easily just cross in to nepal without noticing the Indian Custom and Immigration "Stand". But they will kindly wave at you to turn back haha.
The visas in septmber cost 100usd for three months or 40 dollar for one moth.
the carnet takes about 10 minutes but , make sure you have copies of the carnet and the passports , they will also ask you to take a copy of the visa you just got .
if you have time check out Lumbini its about 10-15km from the border Buddhas Birth place.
From here you have to choices, either take the Siddharta Highway to Pokhara about 180km, very nice mountain road. or head for katmandu.
Pokhara is very relaxed and good place to chill and every one is doing the Annapurna circuit or you can drive halfway up and back again ,there is a new track that goes almost to Mukhtinat .
Kathmandu is crazy , visit Thamel but dont stay there to long its a "sh....t hole" , find a place outside of thamel to stay, there are tons of markets and other sights in Kathamandu. If you fancy trekking make sure you find a realible source !!! 90% of the companies are scammers and just want your money. The other 10% are hard to find without recommendation. If you can afford it visit Tibet by land (permit cost about 4000usd for a motorcycle)
By the way you cannot cross into bangladesh from nepal as it does not border the country. I believe youll need permits to enter the north east provinces of India .
good luck and have fun