I was on an F1 visa in the U.S since July 25, 2013.(thus, I was an exempt individual for 5 calendar years) My H1B was approved on October 1, 2017. My wife is on H4 visa now. I wish to opt for the First Year choice, as I am confident of passing the Substantial presence test in 2018. What will be the starting date of my Residency in 2017?
If you use the First-Year Choice by itself, you would be a dual-status alien, resident starting October 1, and non-resident before that.
According to the Publication 519 discussion on First-Year Choice,
If you make the first-year choice, your residency starting date for  is the first day of the earliest 31-day period (described in (1) above) that you use to qualify for the choice.
This 31-day period is a period of time you were present in the US for 31 days in 2017, and for days of presence in the US, you do not count days you were an exempt individual (which you were during your time on F1 in 2017), so the first 31-day period you qualify for the First-Year Choice under is October 2017 (or possibly later).
Note that becoming dual-status doesn't really add that many benefits compared to being a nonresident alien for all of 2017 (which you would be if you didn't use the First-Year Choice) -- like nonresident aliens, dual-status aliens cannot file jointly (so you must still file as Married Filing Separately for both the resident and nonresident parts), and like nonresident aliens, dual-status aliens cannot take the standard deduction.
Since you are married, there is another option. Once you take the First-Year Choice, making you a resident alien for the last part of the year, you can then use either the Choosing Resident Alien Status (if both you and your wife are resident aliens for the last part of the year, e.g. if both you and her took the First-Year Choice) or the Nonresident Spouse Treated As Resident (if only you were a resident alien at the end of the year, but your wife wasn't, e.g. if she didn't take the First-Year Choice or wasn't in the US at all) choice, which will make both you and your wife resident aliens for all of 2017, and you are required to file as Married Filing Jointly. As resident aliens, you can take advantage of things like the standard deduction. Note that depending on the particular tax treaty with your home country, treating yourself as a resident alien may or may not make you ineligible for certain tax treaty benefits; and treating yourself as a resident alien might also make your foreign income subject to US taxation.