A power strip is just dumb conductors, with insulation round them, with connectors. You use it simply to connect things conveniently, several to a socket, or further away than their power cord would permit.
If the connectors are suitable, then a 240V power strip is usable in 120v and 240v land, but due to the limitations of the insulation, a 120v power strip should only be used on 120v.
Follow the maximum current guidelines, not the power, if you use a power strip at a different voltage. It's the current that heats the conductors and heats the connectors.
If the appliance you want to use is rated at a different voltage to what's coming out of the wall, then you will need a voltage converter, regardless of whether you're using a power strip anywhere to help connect things.
Many modern electronic appliances have a 'universal' input voltage range, typically 90v to 265v, these can be used at 120v or 240v without modification and without a voltage converter.
Given the size and cost of voltage converters, it's almost always best to leave appliances in their 'right voltage' country, and buy a new one for your new location.