I have lived in San Diego for about 20 years.
A good number of rentals are available, but depending on where you are moving from and which neighborhood you select, they may be relatively expensive.
There are many great neighborhoods with contrasting characters and both good and bad points. For what it's worth, the big gay neighborhood is Hillcrest (but it is pricey). The neighboring area, Northpark, is fairly gay and less expensive. I lived there fore years and enjoyed it. The Downtown area is urban, trendy, and crowded, but not very gay. It also attracts a lot of out-of-towners and tourists on weekend evenings and is expensive.
There are many gyms. A popular chain throughout the city and surrounding area is 24 Hour Fitness.
There's a ton of stuff to do, but what will attract you depends on your interests. If you're the outdoor type, there are beaches and hiking trails and much more. If you're into culture it is hardly as intense as some other major U.S. cities, but there are some good museums, a symphony orchestra, opera company, theater, etc.
The zoo, of course, is world famous. There are also many touristy places, such as SeaWorld, which are not my personal favorites.
The weather, of course, is fabulous if you like a Mediterranean climate. If you arrive in June it will be summer. At this time of year, typical daytime highs will be in the mid-70s (but cooler if you live very close to the water or warmer if you live inland) and it will very rarely rain.
If you have more specific questions, post them.