Your swing should go almost entirely forward, with the string bed vertical or slightly closed (facing downward). Because the ball angles up from the bounce into your string bed, it will want to angle up from your racquet face, even if that racquet face is closed by a few degrees. Any upward tilt in your racquet face would make the ball pop up quite high, and given that you'll usually be hitting half volleys somewhere around your own service line, popping up a high sitter for your opponent is the last thing you want to do.
When you hit a half volley by simply moving your racquet forward, what kind of spin, if any, do you think you'll create?
Remember, the ball is coming up across the string bed. That has the same effect, in terms of spin, as the string bed coming down across the ball. Yes, the ball will have backspin, even without your swinging downward at all.
You can add extra backspin to the ball by hitting your half volley with a slight downward chop. This can create an exceptional half-volley drop shot, but it's not easy to do consistently, and a well-softened straight-forward motion can produce a very good drop shot more reliably.
Hitting topspin on a half volley is very difficult and rarely worth the attempt, because the ball is too low to get the racquet below it to brush upward, and a higher bounce due to topspin would usually make your low ball easier for your opponent to hit.
Half volleys are fun to hit, and some players manage to make them fairly aggressive shots, but generally, they're defensive. If you have time to move farther forward so that you can volley, you'll usually be able to be more aggressive. Or, if you anticipate the ball's depth well enough and hang back a step, you can set up for a full groundstroke on what would otherwise have been a half volley.
You might not opt to hit half volleys very often in a match, but you should choose to practice them in your lessons and drills. It takes a fairly good feeder to place the ball at the right depth for a half volley, so if you've never really worked on half volleys, a short lesson might be the best start. Once you get the feel for the shot, a fun drill you can try with a friend is to each stand in the middle of a service box and volley or half volley back and forth, trying to get the ball to land as close to the other's feet as possible.
