If you ONLY get published in tiny little places, people are going to look at your bibliography and wonder what's up with that. Is your work just not good enough for bigger publications? Or do you have absolutely no faith in yourself?
Go on submission grinder and work your way down. Make sure to keep track of who you submit to, and when, and which piece, and what the "if you don't hear after x days, then consider it a no" and make a note of it somewhere. So once they say no, you can check that off and move on to the next one.
Some places allow "simultaneous submissions" which means multiple people can be looking at it at once. But once someone says yes, and you accept, you're supposed to tell everyone else so they can remove your piece from their considering pile. Places that don't accept sim subs have a decent reason to, because there would be offers they'd make and then they'd be told "oh actually someone else said yes already, sorry!" and it wastes their time. A lot of the big markets don't accept sim subs because this happened too much.
Also, some places don't accept "multiple submissions," which means you have more than one piece in their consideration pile. Or maybe they do accept multiples, but only with poetry. Or they have multiple submission windows each year, and you can do 1 per window (and if you got nothing from the last window, consider it a no). This is why tracking everything in submission grinder or a spreadsheet (or both!) is so important, so your wires don't get crossed and you accidentally sim sub or multi sub.
If you go biggest markets down, and someone says no, well, nothing's changed on your end! Nothing lost, let's go onto the next thing. What you should probably prioritize, though, would be anthologies, since those are only open the once, while big magazines are always going to have another submission window later.