What to look for in a good beginner's sewing machine?
Most entry-level sewing machines should do fine for sewing apparel, and can even  handle a few thicknesses of denim (reviews are good for this sort of info). I would recommend going for a well-known brand rather than e.g. a vintage secondhand machine. That way if you have any issues, it will be easier to get support at a local sewing shop, from the manufacturer, when buying parts, etc.

Basic features and stitches:


Running stitch
Zig zag
Adjustable tension and stitch length
Backstitch function
Zipper foot
Maintenance kit (brushes, tweezers, screwdriver, etc.)


These should be standard for any modern machine. You probably will want a buttonhole function and a rolled hem presser foot (huge time-savers IMO). Other specialty presser feet (clear, teflon, etc.) can typically be bought aftermarket - another benefit of buying a well-known-brand, but these should get you through most of your projects.

Once you've brought it home, don't forget to read the manual and give it a cleaning regularly! It's also a good idea to keep it covered when not in use, to prevent dust buildup - some machines come with a cover or case, but you could always make your own.


  With the machine what else would be needed (other than fabric)?


You'll also need:


Thread (in spools, or cones if you have a cone holder - cones are often cheaper per yard)
Empty bobbins
Needles (ballpoint for knit, universal for most everything else; the machine will likely come with a few, but you'll need to replace them as they get dull)


And some general sewing tools, which you probably already have some of if you've been hand-sewing:


Fabric scissors
Seam ripper
Pins
Chalk or disappearing marker for transferring pattern markings
Hand-sewing needles for finishing touches
Iron (ok, it's not strictly necessary, but it will make your results look much more professional if you press everything)


There are many other sewing accessories out there to play with, but those are the important ones.

One last note - steer clear of "hand-held" or "travel" sewing machines. These are small and cheap, typically intended for quick repairs. You don't have to spend a fortune (the model I have is currently $140 on Amazon), but going too cheap and ending up with a finicky, flimsy machine might turn you off from sewing altogether!