I think landlords are usually willing to wait up to one month (assuming you will sign a lease of at least one year) whe you have decided on a place. You will need that time to negotiate the lease, for the landlord to carry out any works you requested, for you to produce references, etc. anyway.
Unless you are renting a high-rent place (say HKD50k or above), don't bother asking for a rent free period. It is much easier to agree on a starting date, then sign the lease based on that date, and just arrange the move on the date you take over the apartment.
And do not stop paying rent on your old place as wtbhotia suggested unless you have VERY good reason to (say the landlord is KNOWN for withholding deposit, which if that is the case why are you renting from him if you knew?). It is against your current lease agreement and is just not gentlemanly.