1. As stated, the sale and the purchase are unrelated, how you spend the money from one does not change the tax on the other.
2. You can't do a 1031 exchange to postpone the tax on the sale or purchase of a personal home.
3. If you sell home #1 within 3 years, you can still use the personal exclusion of gain on sale of your home. To use the exclusion, you must have owned the home at least 2 years, and lived in it as your main home for at least 2 year (730 days) out of the 5 years immediately prior to the closing date.
The exclusion is $250,000 or $500,000 if married filing jointly. So if you sell home #1 in exactly 3 years or less from moving out, you can exclude tax on the first $250,000 of gain, the rest is taxable. (You will also pay depreciation recapture tax on the depreciation you took or could have taken on the home while it was a rental.)
If you miss the 3 year window by even one day, then all the capital gains from sale of home #1 will be taxable. So you will have to have a conversation with yourself around the 2 year mark, is renting profitable enough that you are willing to rent long term and forego the capital gains exclusion, or would you rather sell while you still have the exclusion, and invest the money somewhere else.
You may also want to have your lease contract reviewed by a real estate agent or attorney to make sure there is nothing in the contract that would get in the way of selling the home if you decide to put it on the market while you have a tenant.
