I think a great used machine with just the basic stitches is a good start. I recently gave my niece an older Viking that had very basics , non- computerized.. and its perfect for her needs.
Check the local Sew and Vac or dealers as they take in great used machines and can offer some kind of warrenty. Plus you can test drive it prior to purchase , the cheaper machines sold at Jo'Anns or Walmart in the box , do not allow you to test drive.
I tought several beginners who pruchased very inexpensive machines and had lots of frustration , but when they sewed on one of my older machines they loved sewing.
I found one of the biggest issues with younger beginner sewers is they seem to want to pull the fabric through the machine . I make them use scrap fabric and sit back in the chair and see what happens when they let the machine just take in the fabric. I tell them they are doing it incorrectly if there hands are more than 1 - 2 inches in front of the needle. Once they really "get"that the machine will take in the fabric , the issue of getting finger in the needle is lessened.