How old is the battery? If you don't know for sure, then yeah, just replace it. I got the AGM type for not too much at Pep Boys.
Doesn't sound like the starter, since it turned over and started!
Sorry if this is a dumb question, but are you using the choke? Also, this bike has a vacuum activated petcock, so it won't run unless there is sufficient engine vacuum to the petcock to keep the fuel flowing. Try putting the petcock in the "prime" position (IIRC) for a bit before you start it up to make sure you are getting enough gas. Once running, return it to the normal position (RUN?).
Maybe another dumb question, but are you sure there is enough gas in the tank? When I was a total newbie, I watched the previous owner pour some gas into the tank and start the bike. I made the deal, and then went to get plates for the bike. When I got back and tried starting the bike, it cranked but wouldn't start. I tried and tried, but nothing. Turns out that the jerk PO had drained the gas back out after the deal was done and I had left.
Sorry if these questions seem silly, but some newbies to bikes just don't know these things. I know that when I bought my CB550 all those many years ago, I really didn't know how to operate a choke or a petcock, and then when I bought my Seca II, the vacuum petcock was totally different.
So I finished charging the battery with my new (waterproof!) Battery Tender, and nothing blew up in my face or scalded my eyeballs out. Tried to start it up again. It turned over and started...but when I gave it some throttle it died instantly. I tried it again and I had to hold the start button down after it turned over, but had to keep it held while the engine was half-idling, half-still trying to get a spark, apparently. So, dead battery, and I'll have to shell out some cash and a ride to the nearest Advance Auto Parts for a new one...unless you guys think it's something else.
God, I hope it's not the starter...