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Bayonne is nice, and the A&P near the bridge is a teriffic store; the Stop & Shop in Jersey City is also. But, if it were me, I'd go for Harrison (since I grew up in Kearny) with my grandmother in Harrison. It's a teriffic little town with everything nearby - Path trains can get you into Manhattan in no time, the turnpike and parkway as someone else mention; ironbound has gotten a face-lift - and shopping in Newark, Kearny, Paramus, Livingston Mall, Raritan can't be beat.
I drive over (from Staten Island) to Bayonne and Kearny and Newark often to shop. I would love to live there but my husband works across the street from our house here on SI so there's no commute time - I just travel to NJ to shop.