Once you choose where you'll cross I-20, the rest of the route is simple.
Moving from west of Grant Park to east: Hill St and Cherokee Ave are fine. Boulevard is busier but okay; it's the most direct. I don't know Bill Kennedy Way. Moreland is pretty busy. Maynard Terrace and Clifton St are fine. I also go north-south across Memorial Dr asap because it's busy.
North of Memorial Dr to Emory you have many residential options and you'll enjoy taking different routes. Look at the topo maps and you'll see steeper (though always short) grades on Clifton Rd, etc. Euclid and Whitefoord/Oakdale are two obvious and well-traveled northerly options for you.
The road surfaces are fine -- typical urban asphalt with a share of cracks and small potholes here and there. As for security, you might be less comfortable in secluded areas after dark, but you don't have to be in secluded areas to ride comfortably from Grant Park even in the dark.
Here are three quick options at RideWithGPS:
Grant Park Emory 1,
Grant Park Emory 2, and
Grant Park Emory 3. "2" might be about what you've mapped.
At RideWithGPS turn on the "Bike Paths" option in the top right corner of the map. I find those indicators to be pretty reliable.
And look at the
Strava Heat Map for a history of cycling recorded in that part of town -- it's another indicator of the roads cyclists choose.
