I'm making a serving tray out of five pieces of pine. Will wood glue be enough to hold the tray together? If not, what should I do instead?
Wood glue will do just fine. If the surfaces are properly prepared, a glue joint can be stronger than the wood. Depending on what you want to use the tray for, you might want to consider one of the more water-resistant formulations, but even basic wood glue should do the job, since you don't plan to soak this.
In short: Yes. But...
While glue alone is a perfectly good joint if done properly (you will most likely rather totally anihilate the pine wood than break the glue!) working with "just glue" can be a quite unrewarding experience.
For one thing, glue needs to be clamped, and the wood can get quite slippery with glue on it. The idea of clamping 5 straight, gluey boards together brings up memories of a lunch scene in a Julia Roberts movie: Slippery little bastards.
You can clamp boards on top of and below the to-be-glued boards, but then you should use plastic wrap or something similar as separator. Because otherwise, surprise, with those little drops creeping out of the joint slits everywhere, you will also glue the guides onto the work piece. Very annoying.
If you have any chance of getting a few straight, precise holes drilled, please consider two dowels in every joint. These will not only increase stability, but they also make clamping a lot more of a fun, rewarding experience. With two dowels in place, no matter how slippery it gets, the board has nowhere to go.
Also, note that there are many kinds of wood glues. The kind most people think of when saying "wood glue" is not water resistant in any way, and the type commonly sold as "waterproof" in DIY markets is nowhere near waterproof either, but merely resists short, occasional exposure.
On a serving board, exposure may be quite a bit more than "occasional", not only because of spilling, but also because of cleaning, and keeping it in the kitchen where humidity is often above average.
It probably makes sense to invest in a real waterproof glue (like PU).
 Well, that's not 100% true, the whole piece can, and often will, still warp when you clamp it. But at least the individual boards won't slide off in every direction, which is a huge annoyance. Either balancing force with clamps on either side, or restraining the piece with a (plastic covered) board will do the trick for the warping.