the sensitivity of headphones' drivers decides if a set of cans required an amp or not...
first of all... a amp (in your case, portable amp) is a device that runs on battery, by using the power of the battery, it adds more output to driver some bigger/less-sensitive headphones
if you want your set-up to be portable, im pretty you will not want to carry an amp and exta cable around. so look for some sensitive headphones is a start
from reading your post, i see that you want a set of nice-sounding cans under $70 with great bass and easy to drive....
i can not think of better headphones then Sony MDR-V6.
please note that the cans i recommand are "V6", not V600 which you can find at stores (V600 are crap, but i wont go into that). Sony MDR-V6 are were first released around 1980, but they sound soo good that even today they are being highly recommanded for good sounding headphones..
V6 are very easy to drive, they even sound great with low-output portables. they have very high hights(may be kinda harsh sometimes), very smooth midrange and amazing deep bass (not loud, but DEEP). i own a pair and i love them
used V6s pops up at ebay from time to time, brand new ones are kinda hard to find since they are very old, but it's a lot easier to find new Sony MDR-7506s. lots and lots of people agree V6 and 7506 are exact samething (7506 came out after v6s had been discontinued and added gold plug instead V6's silver). they look exactly the same except for the sticker on the earcups, and from my personal experance they sound exactly the same (also agreed by many others)
highly recommanded, Sony MDR-V6 (MDR-7506)