If you go to the LGA and JFK websites, they give detailed ground transportation information with all the options. Considerations are number of people, amount of luggage, time of day, cost. Here are pages to start with:
https://laguardiaairport.com/to-from-airport/by-taxi
https://laguardiaairport.com/to-from-airport/public-transportation
https://www.jfkairport.com/to-from-airport/taxi-car-and-van-service
https://www.jfkairport.com/to-from-airport/public-transportation
Since you will be dealing with two separate tickets and airlines at two different airports on your return to the US, you need to build in a really long layover as there is no telling if your first flight will be on time, how long it will take to get through passport control and customs and to collect luggage, and most importantly, how long it will take to drive between the airports. Depending on the hour, the traffic between the airports can be terrible. And of course, you will need to clear security again. The airlines generally recommend arriving at the airport at least two hours before your flight. Personally I would only attempt this if I stayed overnight at a hotel near the second airport.
For transportation within Manhattan while sightseeing, use the subway, buses and taxis. You buy a Metrocard for the subway (or bus) at a machine inside each subway station. The first card will have a surcharge to issue it. After that you can reload the card with more dollars as you need it. Apparently you now can also just tap a credit/debit card to pay your fare (this is new!). Here is info on fares:
https://new.mta.info/fares
Here’s more info on how to ride the subway or bus:
https://new.mta.info/guides
Have a fun trip!