I want to upgrade my cockpit so that I have a longer reach. I currently use it as my city bike. Its an old vintage HR bates frame. Currently the stem is a little short. I want to change the bars - flats, curlies or something else? What brakes are recommended?
Be aware, if you aren't already, that installing a longer stem will change the steering response - you will need to put in more effort to steer the same amount.
Another way of increasing reach is to change the position of your saddle. If your saddle has rails, then usually you loosen a bolt under the saddle and slide your saddle backwards.
Also, seatposts have a varying amount of layback/setback (the amount by which the saddle is offset), so you could increase reach by finding a seatpost with more layback.
If you don't want to change your seated position, then to counteract the effect of lengthening the stem, you can increase the width of the bars. I'd suggest remaining with flat bars for city riding, unless you want a more aero position. If you stay with flat bars then you can keep the brakes you have, and maybe refresh the cables and blocks for increased performance.
Another reason for sticking with flat bars is that the bike looks fantastic as it is - it would be a shame to change too much about it...
One great way to increase the reach of your cockpit would be to change your handlebars. With any changes, it is best to do things one at a time and ride on it to get a feel for the new fit. It is cheaper to work with a shop that has components to switch in and out so you can feel the difference before spending loads of cash on new components (this is especially true with stems!)
If you keep your current stem, be sure to measure the diameter of the bar at the stem (bar clamp diameter.) Another consideration is the diameter of the brake lever clamps. Here is more information on sizing specifications. Additionally, your brakes are single-pivot sidepulls and require short pull brake levers (typical for road bikes.)
One bar style that would fit with the aesthetic of a bike like this is bullhorns:
These would maintain the flat portion allowing you to keep your current brake levers, but would give you a more extended position on the horns (similar to riding the hoods of a drop bar or a flat bar with bar ends.) There are bar-end levers that work quite well with these bars if you so choose. Alternatively, you could experiment with adding bar ends on your current bar.
Drop bars (curly in your question?) come in a few different shapes and widths - some are more comfortable in the drops and others more comfortable on the hoods of your brakes. This will likely require a brake lever upgrade, but such is life. From top to bottom: classic road, randonneur, ergonomic, track:
There are loads more options in the world of handlebars that will stretch you out (and more that will sit you up.) I didn't mention aero bars because they'd look out of place on a classy old steel frame, but they'd definitely work.