Salt Lake International Airport is the best option. Provo airport is a budget friendly alternative (note: there are limited flight options because this is a smaller airport).
Lyft and Uber are available, this option is highly recommended. As an alternative you could rent a car or book with a private transportation service https://slcairport.com/parking-and-transportation/ground-transportation/
June is a very sunny month in Utah with little rainfall. You can expect daytime temperatures to reach the mid 80s to low 90s (Fahrenheit). Nights will be warm, with average lows in the 50s Fahrenheit.