I've never been quite able to figure out how groups work in Blender (at least not since 2.5/6x). Is it possible to group several objects and then manipulate them (scale, rotate, translate) as one?
Groups in itself are not that useful. You cannot directly manipulate a group of objects as one object. Although groups make it easier to select objects with Shift + G.
The real power of a group comes from instancing and dynamic linking. Which I will explain:
Once you've created a group, you can instantiate this group elsewhere in the scene by
Group Instance. This saves you the trouble of creating literal duplicates of your objects. Consider a city scene, I might have a number of objects that makes up a car model. By grouping these objects into a group, I can place a car group instance anywhere in the scene with very little overhead.
In many ways, dynamic linking is an extension of instancing. The only difference is that we are using a group from another blend file(often called a library). Dynamic linking is a very useful feature in a large production, because it allows larger scenes to be broken up in smaller chunks. (managed by different people)
Yes, it is possible. As Mike Pan wrote, making a group itself is not useful to transform several objects as one object, but if you make a group in a separate scene (or a separate blend file) and add an instance of the group to the current scene, you can transform these objects as one object. That is, you do not transform a group as one object, but you transform a group instance as one object.
But it may be more convenient to use parenting than a group instance if the sole purpose is to transform several objects as one. For example, add an empty and set it as the parent of the objects you want to treat as one. Then transforming the parent will transform its children together. The difference from instantiating a group is that it is easy also to edit individual objects if you want to, because the individual objects are also in the same scene.
Grouping in Blender doesn't work like it does in many other apps. See Mike Pan's answer.
To "group several objects and then manipulate them (scale, rotate, translate) as one", Just select them by selecting one of the grouped objects an pressing ShiftG> Group, then manipulate them.
Note the different Pivot point settings will change the way this behaves:
Active element: Transform around the active (last selected, with a yellow outline) object center.
Median point: Transform around the center of all the selected object centers. (This is most likely the behavior you are looking for, treating the selected objects as one object)
Individual origins: Transform each object separately around it's own center origin.
3D cursor: Transform the selected objects around the 3D cursor.
Bounding box center: Transform selected objects around the center of the bounds of all the selected objects.
To "group" object as in "the other apps" you can use parenting. Parent them to an empty (CtrlP). It can be practical, but it is not an orthodox use and can cause some problems, for example in animation.
You can also join multiple objects to make a single new object (CtrlJ), but keep in mind it can be unpleasant to separate them later if you connect them with edges or faces, otherwise you can use P> By loose parts.