I'd start off just riding your bike and noticing what you like (and don't like about your position). Mostly I'd worry about the bars being too low and not having enough options (get low to get out of the wind, sit up straight when your back or neck needs a break, and on and on...). The important thing is that you're comfortable – and as long as the bike comes reasonably close to fitting you you'll be able to make adjustments that will make it very comfortable.
But back to your question. There are three general bar options:
- Flat bars like you've got and their variants (mustache bars, albatross bars, hershey bars, all sorts of fun to be had here). If you don't ride hard and like an upright posture, this would be where to start looking if you need more variety in your position.
- Trekking or butterfly bars, these are basically flat bars with a larger range of positions. The forward position is a way to "drop down" a bit to avoid wind drag.
- Dropped or road bars. These are my favorites. I like the range of positions and I'm used to them. But, really big but, they are close to useless if you can't get them up high enough. I run mine with the top of the bars about level with my seat, so riding on the tops of the bars my posture is pretty much the same as it would be with flat bars. However, I can also lean a bit forward, riding with my hands on the hoods of the brake levers, or if there's a headwind or I'm trying to look cool and fast I can drop down onto the bottom of the bars. But most of the time I ride on the top of the bars or on the hoods. So I think you don't need to be in any rush to buy new bars.
One last thing, it isn't the length of the ride that dictates when aerodynamics comes into play, it's more the speed (either your speed or the speed of a headwind). At 15-20 kph I don't feel like I'm working too hard against the wind, but up around 25 kph I definitely notice it (but not enough to make this old man want to spend much time of the drops), up past 30 kph the drops are looking very attractive.