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Yes, standard print is 300 dpi. 72 is the standard res of older monitors, bear in mind that higher res monitors are now common e.g. Retina display Macs, smartphone etc.
Changing the resolution on an image will only affect its quality if you also force it to change size in relative to the resolution. For example increasing the size of a 72dpi image will reduce its quality quite rapidly, increasing the resolution will not help.
If I print a 72dpi image to retain quality I need to print it smaller than its size at 72dpi because I need to get it as close to 300dpi as possible. (Having said that in print productions you can get away with some image being as low as 180ish dpi even for publications without the average reader noticing.
These days also a 72dpi image will appear degraded on mostnewer devices when viewed at 100% (full size).
Ultimately it depends on the images usage. Default rule in my world (graphic/web design) the larger the resolution and size (dimensions) of the original the better as this will be more flexible later.
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