Summarize:

Until you are able to get the cat to the veterinarian for a checkup, it is best to handle the cat while you wear protective clothing. Wear thick gloves, a long-sleeved shirt and long pants to minimize the chance of getting scratched or bitten. After you've been petting the cat with a spoon for a while, try sneaking your hand under the spoon and petting the cat.  Only pet the cat's shoulders and head. Don't get anywhere near its underside. A cat will be very defensive if it feels threatened. Only when your cat fully trusts you will you be able to pet its belly. Use a towel or blanket to pick up the cat. Do this after you've pet the cat a few times. Choose a time when the cat seems calm and relaxed.  It can take a long time to get to this stage. It really depends on the cat. Some cats will never be truly tamed enough to pick up. If the cat struggles when you hold it, let it go. You may end up getting scratched or bitten. You might also undo all your hard work getting to this point.
Wear protective clothing. Use your hand to pet the cat. Try picking up the cat.