New Job Offer - Hand in Notice + Confirming Start Date - Process/Sequence

Hi

I've just verbally accepted a new job offer and as such will soon have to give notice to my current employer once I have my new contract paperwork to hand...

I've been sent an electronic copy of my new contract to perouse and at the start date there is a "available for employment from" field, and the recruitment agent has also asked twice about when I'll be available.

I am contractually on 3 month notice with my employer, so clearly first stage is to hand in my notice.... but from there, what do I do:

a) when/how shall I ask my current employer about reducing my notice period? - I work as a consultant so there is my internal boss and assignment bosses - all of whom will have an opinion
b) what shall i declare as my "available to start from" date, or is it ok to t say thats "subject to agreement"

I am just getting a bit "nervous" by the amount of attention this stage of the process is getting, although I understand its the next (and key) step!!! - Hoping wont have to serve the 3 months as thats a long time to wait, that wouldnt cause any issues would it!? - The new place are aware I am on 3 months though we've spoken that notice periods can be negotiable sometimes.....

thanks in advance