Both the AT Pro and the 'carrot' are proven good products from Garrett, so your choices are sound.
If you are shopping for a good deal, looking around is not a bad idea, you may find a deal out there.
If I was going to jump into detecting, these are some factors I might consider...Amazon/Ebay may not offer 'bundle' deals where as a detector dealer may. A dealer will offer knowledge or advice with operation of your purchases where Amazon
or Ebay surely won't. A purchase in person let's you try before you buy option where mail/ delivery does not. A good and easy return policy is always a factor to consider in case things go that way.
These products like many are mass produced and there are lots out there in the marketplace, used equipment can just as reliable as new and come deeply discounted from retail prices, so that is also an option, but you may be stuck with somebody's old headache , so consider that as well.
Whichever way you go, and detecting proves to be an activity you become involved in, I'm certain you will find it rewarding and will give you lots more options of adding to your coin collecting because it has for me.
Goodluck out there..theres plenty of stuff just waiting for you to find it!