First and foremost concern should be the ability to fix flats on the road, so you need to buy flat tools like Soma Steel Core tire levers, patch kit that includes glue, sandpaper, and patches, alcohol pads, a pump and a spare tube. If you don't know how to fix a flat just go to YouTube and type in "how to fix a bicycle flat tire" and watch several of them then practice over and over on your bike, you don't want to learn on the road! Of course in order to carry that stuff you'll need a saddle bag, I like the Topeak Aero Wedge bag, they have several sizes and they expand. Pump wise I like the Lezyne Road Drive, they have 3 sizes, small, medium, and large, get only the large one it will pump tires easier; I don't like cheap pumps for one they will never get close to recommended road tire pressures, and two they break fast.
Next line of importance would be a multi tool BUT only if you have mechanical abilities to fix a bike, if you are mechanically backwards there is no sense in buying a multi tool if you're never going to use it. I happen to like the Park MTB 3.2 multi tool because it has everything you could ever need, but there are quite a few multi tools on the market.
Then you need to get water bottle cages along with water bottles for longer rides.
By the time you start riding longer distance requiring hydration you'll need padded cycling shorts, don't go cheap on those and read the reviews before you buy. You also should get a jersey, but you don't need expensive jerseys, any $15 or less jersey will work just as good as $150 ones; this is also true with socks, get cheap 100% polyester socks as cheap as you can find.
Next would be a rear tail light, one that is very bright. These can be used in daylight too.
Headlight maybe if you ride at night or live in an area where is rains and or has fog.
Pedal wise you can wait quite a while without ever needing them, but there are a lot of good pedals on the market, I like Speedplay and Shimano, but there are a ton more that are also good pedals. Of course when you decide to get pedals you'll need to buy shoes and shoes aren't cheap, cheap shoes could flex to much and hurt your feet on long rides. You should have a pro fit done and let the bike shop determine what pedals and shoes to get, be careful of pro fits can be a ripoff, so ask around to find out who has the best one in your city.