The most common way in the construction room to assemble wood projects is to use wood glue. Wood glue is extremely strong, non-toxic, water soluble (clean up with water). However, it takes up to 24 hours to cure to full strength. Therefore, it is important you have a system to hold your pieces firmly together while the glue dries. Clamping is generally the best solution.
Good glue-up means using not too much glue and not too little. You can use your fingers or a glue brush (silver handled) to apply a thin layer of glue - typically to the end grain if using a butt joint. When you squeeze your pieces together you should find SOME glue oozing out of the joint - but not too much. Make sure to clean excess before it dries!
If you have a project drying with clamps, do NOT put those clamps in the cupboards. Leave them on the back counters to dry.