Lot of good advice here already.
I'd go along with the buy the best lens you can afford advice. However, lenses can get really expensive really fast... but in general, since you'll be photographing your kid (moves around a lot) and may eventually be in places with poor lighting (birthday party/candle-blowing, recitals, etc), you'll want a fast lens. These lenses will have a lower f number (wider aperture). However, it will also give you a smaller depth of field (less of the photo is sharp and more has that blurry background look).
I'd definitely second the recommendation of getting a book. I TA-ed photography classes and used this
book.
Better yet, I'd look into taking an intro photography class at a local college or university. You'll learn a lot there and hopefully get critiques and such in class.
Lastly, you may want to also look into investing in editing software (Photoshop, Lightroom, Aperture, or Photoshop Elements if you are on a budget) and learn how to use them (they all have steep learning curves as well). Using them is like developing your photos in a digital darkroom and can really improve your photos.
I'd say taking great photos involve good equipment, good technique/technical knowledge/good eye (can be developed), and good editing. If you can cover those three areas, you'd come out far ahead of many other people's photography
Oh one more thing... in terms of mirrorless vs DSLRs... both will probably do well with what you'd want. However, mirrorless cameras will have a smaller selection of quality lens and may not have quite the same image quality as an SLR (though make no mistake, they can still take great pictures). With the DSLR though, you're giving up compactness and weight. If you carry a camera all day (plus lenses), it can get really tiring and heavy!
With mirrorless, I've enjoyed using Sony, Samsung, and Panasonic models. With DSLRs, you can't go wrong with Nikon or Canon. If anybody says Nikon is FAR better than Canon (or vice versa), they're usually just fanboys along the lines of iPhone vs Android fanboys here. Go to a store, hold them, try them out, and pick the one that you are most comfortable with. And look if either system may have lenses you may be potentially interested in.
Hope that helps and happy (future) shooting!