I'm thinking of submitting a short story to a small (and probably not well-known) serial anthology publication.
For an author who's never been published in any way, shape, or form, should I be wary of the rights I'm asked to give up, or should I just concentrate on getting work published as a first step, regardless of what's asked.
For example: first Canadian rights, non-exclusive anthology rights, first online rights.
How prolific are you?
I'm pretty prolific, so I honestly don't worry too much about what rights I'm selling - I try to work with good publishers, and then assume that they'll do a good job selling the work associated with the rights I've granted them. But that's because it's not that big of a deal if I give rights to someone who doesn't deserve them - I'll just write something else and sell it elsewhere. If you're less prolific, you need to worry more.
Even prolific authors need to worry about some things, like giving publishers rights of first refusal on future works. Anything related to future work, really, negates the 'prolific' exemption, because being prolific in that context just means that you're binding yourself to the publisher for even more work.
But, honestly, I'd say that the examples you give are pretty inevitable if you want the work published. I'm assuming it's a Canadian publisher, so that makes sense, non-exclusive anthology rights are pretty much the minimally binding terms given that you're looking at an anthology, and first online rights just means that your work is going to be published online (at least, this is my understanding - I Am Not A Lawyer).
It's perfectly OK to give up pretty much any rights to the story, as long as you feel you are being justly compensated for them. If I were to make such a sale, as a no-name writer I wouldn't expect to get a large amount of money, but that's OK. Stephen King, on the other hand can reasonably expect a larger sum for his efforts.
If you can't afford a lawyer (it's a short story, so I'm betting you don't have a lot of cash left over for lawyering after you buy Ramen), then read it carefully and make sure it enumerates, with precision, the rights you are giving up, and how and when you are to be paid.
You might also want to try to get some kind of 'reversion' clause on any rights where that makes sense. For instance, with first publication rights, it doesn't make sense to have reversion clause - but if they retain the sole right to reprint the story, you'd like for that right to revert to after some period of inactivity (several years at least).
If you can get it you'll also want some kind of reversion clause for the case where they don't actually ever publish at all. Again, this would be a several year timeout, but if something goes wrong after you've been paid, you eventually want the story to see the light of day.
Have you received a contract? You said a short story. Have you been sent anything by the publisher? What rights are you giving up by just sending the story?
Typically they have to acknowledge that they are going to publish it. Are they?
You might have the horse a mile or so in front of the cart.
If you have a contract then you might need an attorney familiar with the publishing business.
