Depends on what the table is used for. Given the description of wear, let's assume it's a college project work table, that needs to handle anything from homework writing, gaming, to x-acto knife cutting, to light woodworking, painting, etc.
I would suggest hardboard (aka Masonite). It ain't fancy, but it's cheap, widely available in 4 ft x 8 ft sheets (so no cutting required), and suitable for this level of usage.
You could glue it to the existing surface using regular wood glue, however you might think about instead just nailing it in using a lot of small tack nails; this way if the surface becomes too heavily marred, future maintainers could just rip it off and replace it with a new sheet of hardboard.
I use hardboard for surfacing my work tables in my shop. I generally throw a couple coats of polyurethane on top. I suspect you could use pretty much any sealant you wanted on it (or none!) If it's going to be used more for homework/gaming, and less for crafts, a hard coat might make sense.
As it appears the 'wobble' is between the metal leg, the metal bracket, and the bolts provided, I'd say this wobble is due to IKEA's ahem rather wide tolerance specifications.
In other words, sometimes you just get wobble when it comes to IKEA products. :)
I'd suggest using something to solidify the connection between the leg and the plates. JB Weld would be a possible option. Mix it up and put a generous amount between the leg and bracket, then tighten up the bolts. Let it set up for about 24 hours and you should now have solid material between the two parts to shim the wobble away.
Best Answer
The minimum would be diagonal bracing on back and side.
Triangles resist lateral forces much better than rectangles do.