Standard PVA wood workers glue - the yellow stuff, not white Elmer's like you
ate used for grade-school craft projects, applied very liberally and clamped very tightly will create a bond that's stronger than the wood itself.
You will need a lot of clamps to get even pressure across your surface to ensure a good bond, though. The common wood worker's adage of "you can never have too many clamps" is very true, especially for glue ups like this.
I'm not sure if you're planning on having the 2" or 4" side showing, but in both cases, especially the 4" case (applying glue to the 2" face), you'll want clamping cauls to ensure that your boards stay aligned and the surface reasonably flat while you're applying pressure and waiting for the glue to dry. This is a pretty thorough write up on making some cauls.
That was a likely looking candidate from the search results for "wood clamp caul" - there are plenty of other options, so no implication that you need to go to that extreme.
Note the number of clamps used just to make the cauls to give you an idea of the number of clamps you really should use on your table tops.
Other points of note:
- Make sure you look at the grain at the end of each board. Unless you get very lucky with your selection of wood, you'll see that there's a curve in the end grain. Orient your boards so the curves go in opposite directions for each piece. i.e. start with the first piece curve down, then the next piece curve up - keep alternating. Rearrange pieces as necessary if you have some not-so-good faces showing. This will help keep the surface flat over time and resist warping.
- When attaching the skirt boards or other bracing underneath, I'd strongly suggest using table maker's clamps to hold it all together. Wood expands (in all directions, but mostly) across the face of the grain (the 4" face of your 2x4). Table clamps will allow for this wood movement so you don't end up with wood splitting in a couple of years. If you try to screw or glue solidly across the face of the grain, the different expansion directions will cause failure, it's just a matter of time.
- If you have other questions about the woodworking aspects of your project, I'd recommend you stop by Woodworking as there's a lot of expertise over there that may not find your questions over here.
- Ensure you liberally apply glue to each surface to be glued. You'll get squeeze out, but A) that cleans up with a damp sponge, and B) that's better than a dry joint which will fail. (Cover clamps, etc. with waxed paper to keep your tools and work surfaces clean.)
- No, really, you need a LOT of clamps for a table sized glue up and you need to clamp really tightly. No, tighter than that! Keep going... Keep going... You're getting closer...