Here is a neat site with a LOT of how to get started, tips and tricks, etc.http://bicycletouringpro.com/blog/
Re. stoves/cooking stuff vs. not: Of course this is personal, and it's possible to ride across the country without it, but that means you are always facing either cold meals and/or someone else's cooking--not often appealing prospects. One of my favorite things is brewing up a hot cup of java to greet the morning and get to my packing chores. Camped in the middle of nowhere (my preferred location) this is quite impossible without a stove.
These can be quite light. I rode across the USA using only a small alcohol stove built out of two Pepsi cans that I found on the side of the road during a training ride--fantastic little bugger.
If you are cooking for more than one, however, these units are less than optimal. If you have more than one person, check into a Whisperlite International by MSR. Whatever stove you get, you want to make SURE it burns unleaded gasoline well. Finding Coleman style white gas in small quantities on the road is almost impossible. Short tours, however, are less problematic. In that case, white gas, bottled gas stoves that use propane blends, etc. can all be fine. For short tours, I often use a little Bluet bottled gas stove because it's so convenient, quick to light and compact.
Sierra Designs makes a superb one-person tent: the Light Year. There are many others out there. Follow the advice of the other poster about tents--get a good one. Also, crucial for me, is an excellent sleeping pad. Go to a good outdoor store like REI and spend some time lying down on different pads. Generally, thicker is better! I may skimp in other areas, but I splurge when it comes to sleeping pads. For me, an extra pound or so of sleeping pad is worth every ounce. Currently, I'm using Big Agnes air mattress, which is pretty light, about 3" thick, but does require lung power. Anyway, some time spent checking out different models is well worth it. I find the super light, closed-cell Ridgerest/Z-rest type pads unacceptable. Remember, you'll be on it for HOURS at a time. On winter tours, for example, we're in the tent for about 12 hours. Ugh. Gotta have good sleep gear.
Re. bike: Make sure it really fits. In general, you'll probably be more comfortable with a more upright seating posture, and make sure whatever bars you use have multiple hand positions. The saddle is everything. Brooks seem to be the gold standard, but they do take some break in. I ride a recumbent, so this not an issue for me.
Anyway, lots to think about! Enjoy the process. Have a great ride.