Congrats and enjoy!
Like a lot of other people, I've got a small saddle bag with stuff I'd need to fix a flat tire plus a basic multi-tool in case of mechanical emergency. And a helmet.
Here's a quick rundown of a typical saddle bag and its tire-fixing contents:
* A saddle bag like this
Serfas Speed Bag. A "small" one will carry your tire-fixing stuff. Move up to a bigger size and you can probably stuff a small wallet and phone in there, too.
* Spare innertube (because it's quicker and simpler to swap a tube than patch one on the road).
* Vulcanizing patch kit like the ones sold by
Rema or
Park Tool (because it sucks to walk when you get a second flat tire).
* I carry a CO2 inflator in my road bike's saddle bag, but keep a mini pump on my other bikes, like the one I ride with my kids. Topeak, Serfas, and Lezyne make good ones. The right one for you would depend on what size tires you're using. Road bikes with skinny tires need high pressure, which some pumps can't manage. Bikes with fatter tires don't need as much pressure, but need lots of volume, which can take forever to attain using a pump made for high pressure.
* Basic multi-tool like this
Park Tool MT-1. There are a whole bunch of other options in various shapes and sizes. I like the ones I have from Park Tool and Crank Brothers.
Another accessory I use on occasion are a rack and panniers. Sometimes I go riding with my daughters and need to haul some stuff for the trip -- lunches, extra water, ball gloves, medical supplies, etc. I hate wearing a backpack when I ride, so I mounted
this rack and use a set of panniers (bags) similar to
these. They also make it practical to use the bike for errands like trips to fetch a few things from the store.