I rely on Titebond - and you can get it in different strengths but I've always found the "Original" to work just fine. The advice on using clamps is sound. I've literally glued hundreds of feet of wood bonds using it over the years, never had a problem (that wasn't a result of my own errors!).
Any decent hardware store and many craft stores will have it.
Smaller parts still work, just be judicious in the amount you use. Squeeze out onto a throw-away surface and then use a very cheap brush to apply only as much as you need. You want to avoid any drip - and have a damp paper or cloth towel ready to wipe any 'leakage'. Another trick is to wait a few minutes before adhering pieces together to allow the glue to get tacky - but don't wait too long. The bond, once cured, is supposed to be stronger than the wood itself - with any wood glue. You don't get any "re-dos" with wood glue.
I am US based.