Finding a suitable ice axe for mountaineering is easy. Choose a plain looking ice axe with a straight shaft and no rubberized grip. Avoid ultra light axes as they are often weaker and more difficult to use effectively. Avoid tools intended for ice climbing as they perform poorly for general mountaineering. These are no good substitutes for a basic ice axe. Find the proper length by gripping the top of the ice axe like a cane and holding it at your side; the tip of the shaft (aka, the spike) should be at your ankle bone. If in between sizes choose the shorter ax. You should buy a simple ice axe to begin with unless you have a specific reason to buy something else. The Black Diamond Raven is a great example of a basic mountaineering axe. I still use a Grivel I bought back in 2002 for all of my mountaineering. A good ice ax can last a lifetime.
After gaining experience many mountaineers discover their basic ice axe, while great for most things, is not best for some of their climbing. As climbers venture onto steep terrain many prefer shorter axes. A shorter ice axe provides less security in low angle terrain, less than 45°, because they are more difficult get a good self belay with but are generally easier to maneuver on steep terrain, over 60°. These more specialized axes may also have rubber grips, curved handles, and interchangeable picks that improve performance on steep terrain but the rubber grips and curved handles can make self belay more difficult and technical picks for ice climbing perform very poorly for self arrest. At the extreme end of the technical spectrum we have technical ice tools that are designed for steep ice and mixed climbing. While they excel at these tasks they tend to be poor at self belay and self arrest making them unsuitable for general mountaineering.
It can be tempting for a beginner to buy an expensive ice ax with many technical features but this is a mistake. For beginner and intermediate climbs the basic ice axe is right almost every time. A basic ice axe will take you to the summit of every state high point and far beyond. My highest ascent in the Himalayas was done with the basic ice axe I bought for my first mountaineering trip. If you are still unconvinced look at photos of climbers on Mt. Everest, they will be using a basic ice axe for their climb. This is not to say the more technical tools don’t have a place, they do, but they are for experienced climbers who know why they need the advanced features and why the performance compromises are acceptable.
Thank you for reading. If you are new to mountaineering be sure to find a good mentor or consider taking classes to gain proficiency. You will encounter many hazards in the mountains that must be appropriately managed to mitigate risk. An inexperienced mountaineer will be unable to assess many of these hazards. Stay safe and have fun out there.