Florida does not have an income tax, so that part will make it a bit easier.
After you become a resident of FL, you will file NY as a part year resident. As to whether or not you have to pay taxes on all of your income for the year to NY will depend if you meet the Convenience of the Employer rule.
The Convenience of the Employer Rule basically states if you are working remotely for your own convenience, then it is considered NY sourced income. If it is for your employers convenience, then you would not have NY source income for the days you do not work in NY.
The fact that you will be in NY during the summer, if you work during the summer, means you will still have to file a return and claim the income for NY for those days regardless of whether or not you meet the convenience of the Employer Rule as you would be earning it in NY.
Read over the rules in this link to determine if you are working for your convenience or your employers convenience.
If you do meet the test, then you would only pay taxes on the income you earn while inside the state of NY. The rest of the income you earn while in FL would not be taxed at all.
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