if you wanna use the same tires you can only get a rim that is 15". a 15" tire will not fit a 16" rim or any other size rim other than 15". i'm not sure of the width of stock rims but i believe they are either 5.5" or 6" wide

a 16" rim will handle better only if you get a wider rim. like say 16x6.5 or 16x7. then you can go with a wider tire. the wider the tire the better they will handle. it will ride a little rougher but nothing that you will notice. now if you go with a 17"-19" then you will notice a difference

if you go with a 16" rim/tire package you can get them for around $600-$1000 depending on what you get

check out
www.wheelmax.com
www.wheelmart.com
www.tirerack.com (i think they just have tires)
www.groupbuycenter.com

for the rims. i believe the bolt pattern on the 2003 Accords are still the same. so if its a V6 its 5x114 and the I4 is 4x114 (some may have them listed as 114.3 but its the same thing)