Welcome to Arizona!
Sweatwater would be my first recommendation for Tucson, so I am glad you found it.
Another area to the south, just east of Green Valley, is Medera Canyon. It is higher than Tucson so it will be slightly cooler. Stop at the gift shop. There are feeders there which attract a good selection of birds and there are benches for great viewing. You can also hike some of the trails that follow the creek.
http://www.friendsofmaderacanyon.org/
There was an article a while back that the Ostrich Farm was having financial issues and may be closing down. However it was still open about a month ago when I drove by. I would go there sooner rather than later, just in case.
http://www.roostercogburn.com/
Another area close to Tucson is various spots on the drive up Mount Lemmon that borders the north side of Tucson. You start out in the desert and end up in pine country so there are many varieties to see.
TheSanta Cruz Flats sod farms to the north between Picacho Peak and Arizona City off I-10 is popular when things cool down. There are shore birds by the sod farms and one highlight is the Crested Caracara found west of Pciacho Peak.
http://butlersbirdsandthings.blogspot.com/2012/12/for-next-week-or-two-i-am-afraid-i-must.html
If you do not mind driving a little more, then check out the Gilbert Water Ranch southeast of Phoenix. It is the best spot I know of to see a lot of birds with little effort.
http://www.gilbertaz.gov/departments/parks-and-recreation/riparian-preserve-at-water-ranch/birds
Another similar site, but not as mature, is the Veterans Oasis Park a few miles south of Gilbert Water Ranch, in south Chandler.
https://www.chandleraz.gov/content/VETOPMapBrochure.pdf
A great spot during the winter is Whitewater Draw near Tombstone, AZ. It is home to thousands of Sand Hill Cranes and geese, along with more water fowl.
There are some good parks in Tucson for birding and check out the cemeteries with older trees.
A must if you are a Tucson area birder is to get the Tucson Audubon Society book called Finding Birds in Southeast Arizona. It will have details on some of what was mentioned above plus a lot more.
As an FYI, the Tucson Audubon Society is having it's annual festival in August. Parts of it are free to the public and is worthwhile to visit, especially being new to the area.
http://tucsonaudubon.org/what-we-do/festival.html
Here is a great blog from a knowledgeable Tucson area birder. Browse though some of the older posts to find some great locations and details on what can be found. He birds many areas outside of southeast AZ so you will have to look around.
http://gardeningandthespanishway-rohrerbot.blogspot.com/