Use a different size post for gate than the fence

Assuming that they are round metal poles that are generally used for chainlink, the 'corner' poles are what you generally use for a gate. You pretty much use corner poles anytime you terminate a run of any kind, and a gate opening is a termination. The actual gauge isn't going to matter so much as the diameter of the pole, you need to make sure your gate hardware will fit the pole. Since your are going with wood, the extra few bucks for the thicker gauge is probably worth it, the actual diameter should be about the same either way.

Also, make sure you measure the opening correctly before concreting the poles in. Depending on the hardware you use you may need to leave extra space. For pre-fabricated chainlink gates, you need something like 39 inches for a 36 inch gate. Since you are using wood, I assume you are fabricating it yourself, but you'll still want to make sure you leave plenty of room.

How tall are your posts to start with? A 6 foot above ground post should have at least 2 feet in the ground in your area. I would be setting them in concrete so you have a good solid base and good ballast. A metal post in concrete won't rust as long as you dome the top of the concrete so water runs away from the post.

I'll share a clever trick that works well and is very fast. After digging your hole for the post, set one or two bricks in the bottom of the hole to set your post on. You will of course have to adjust for height and plumb the post. Then simply pour an 80 pound bag of Sacrete premixed concrete mix around the post dry, right from the bag. Now just pour apx 2 gallons of water on top of the mix and leave it alone for 24 hours. You will have a solid concrete base for your posts without having to mix concrete. Mixing concrete is good, even better, but this little trick is fast and easy.