What type of wood is recommended for an outdoor jungle gym?
The go-to materials for outdoor construction are cedar and pressure treated wood.

Many prefer cedar, and most kits you see from big box stores and other sources use it. It is generally more expensive, but it is lighter and softer (for drilling and hammering). It will last for a number of years untreated, and even longer if it is pre-treated with penetrating sealer and occasionally retreated.  Be sure to coat the end cuts for maximum benefit.

Pressure treated woods are also very durable and will also last years with little or no maintenance. In the past, the treatments used harsh chemicals (called CCA for short) that were not ideal for skin contact and when cutting the wood, dust precautions were required. More recent forms use less caustic chemicals and may be allowed in your jurisdiction. (Personally I would avoid pressure treated where food or kids are involved).

While all outdoor projects require hardware that is weather resistant, pressure treated woods need specially coated fasteners to avoid a chemical reaction that causes corrosion.

After pricing good quality materials, you may find that kits from manufacturers may be a fairly good value. I recently built a swingset with a fort, slide and climbing wall from a major manufacturer.  It arrived on a 4x8 foot pallet with all cedar, hardware and plastic parts. The instructions were surprisingly good, all parts present and in proper order. Cedar, stainless, and plastic. Even predrilled correctly (one hole at wrong end - no biggy).

A search online will show you these options in both big box stores and directly from manufacturers or their distributors. There are surely units that cost many thousands of dollars, but a significant set can be bought for several hundred.  Many sources offer onsite assembly for a few hundred dollars.  But where would the fun be in that? The look on my grandkids faces made the cost and time in building all worthwhile.