Atami City

I will be stayin in Atami City for a nite. Is there any thing in particular that it's a must-go?

As i'll be making use of the stay in atami as it's more budget as compared to Hakone. In the day, i'll travel to Hakone. So i would like to know if there's anything in Atami which is not to be missed.

The hot springs are Atami's prime attraction. I also very much enjoyed the MOA Museum of Art:
http://www.japan-guide.com/e/e6306.html

is it advisable to spend 3/4 the day at Hakone .. since there's not much attractions in Atami? I stay at Atami as the accomodation rates are more budget.

Yes, why not. And Lake Ashinoko is only a one hour bus ride away. Note, however, that the bus betwen Atami and Hakone is not covered by the Hakone Free Pass.

To take the bus from Atami to Hakone, how do i do so? Can i go by JR ?

How much will the bus costs ?

And i see people mentioning romance car .. wat's tat ?

To take the bus from Atami to Hakone, how do i do so?

There is a direct bus from Atami Station to Hakone-machi and Moto-Hakone at the coast of Lake Ashinoko:
http://www.izuhakone.co.jp/bus/rosen/timemenu/t0006.htm

Can i go by JR ?

Yes, you could also take JR trains to nearby Odawara and Mishima Stations and take a bus or train into Hakone.

How much will the bus costs ?

The bus from Atami to Moto-Hakone costs 1150 yen one way.

And i see people mentioning romance car

That's a limited express train by Odakyu Railways, which operates between Shinjuku and Hakone-Yumoto.

Atami was, and is still some degree, a site popular with group tourists. Nowadays not exactly popular among Japanese.
If you would like to experience real Japanese hot spring experience, I'd rather recommend smaller hot spring towns. There are hundreds of them in Japan.

Can i go by JR ?

Yes, you could also take JR trains to nearby Odawara and Mishima Stations and take a bus or train into Hakone.

I'll be takin JR then, since I've the 7-days pass. How much does the train / bus cost from Odawara station ? And how long will it takes ? I can easily find the bus service to Hakone right ?

For the hot spring places, mind to recommend some .. perhaps in Hakone since i'll be there for 3/4 the day ... Thanks.

Odawara is at the entrance to Hakone. Which train or bus to take from Odawara depends on where in Hakone you wish to go. See the second map on the following page:
http://www.japan-guide.com/e/e5206.html

Here is a list of some public hot spring baths in Hakone:
http://www.japan-guide.com/e/e5209.html

I like the Tenzan, but hear that it can get quite crowded.

Is there any private bath available there ? As it'll be a honeymoon, we would prefer privacy for that :)

Among the baths listed on the before mentioned page, the Rakuyujurin Shizenkan and Yunessun's Mori no yu have private baths. Advance reservations are recommended.

Rakuyujurin Shizenkan and Yunessun's Mori no yu
How do i do the booking ... as i can't find any website or number to call for reservations ...

The Rakuyujurin Shizenkan seems to accept only same-day reservations from 10am. The phone number (not sure whether they are able to deal with inquiries in languages other than Japanese):
0460-82-0265

The place is not easy to find, so make sure that you know your way in order to honor your reservation.

As for the Mori no yu in Yunessun, description of private baths and phone and fax number can be found here:
http://www.yunessun.com/english/mori_no_yu.html