Hi MovinOnUp1919,
I moved away from the area a few years ago but check out Westfield. It's a very nice town in Union County, not far from Elizabeth, with a great downtown shopping district.
Not knowing your price range or what size apartment you're looking for, I did a quick search for apartments below $1,500 in Westfield and found some examples at this Weichert site:
http://www.westfieldnj-rentals.com/h...nt/0-1500.html
Some of them are very near the downtown shopping district, close to restaurants, Trader Joe's, shops, a movie theater, and the NJ Transit train station and bus service is very close by.
Prices are going to be similar for any area that fits your description. You could also try Scotch Plains, Cranford and Clark. They're all in Union County and close to Elizabeth.
NJ.com also has rental listings for all those towns. Just Google the town name + NJ + rentals and look for the link to NJ.com listings.
Good luck at your new job and in your new apartment!