Diameter to breaking strain ratios vary quite a bit between different nylon formulations but generally speaking 0.4mm diameter tends to come in at about 20lb (9Kg) breaking strain and 0.6mm diameter tends to come in at about 35lb (17Kg).
Unless you are throwing big lures for big fish or fishing at close rage into very rocky water, both of these are overkill for shore based lure fishing and will have a serious adverse affect on your casting distance.
Personally, for general lure fishing on the coast, I would look to be using a line of about 15lb (7Kg) breaking strain and with a diameter of about 0.3mm and would probably have a couple of feet (40 - 60cm) of heavier line knotted to the end as a rubbing leader.
Lines of this strength are about right (probably still a little heavy) for a 4000 size Shimano and will have absolutely no problems landing fish to well over 20Kg's from an open beach provided you pay propper attention to using the right knots and correctly setting and using your reals drag system.
Correctly filling your spool with line of about 15lb breaking strain will also see you able to cast at least twice as far as with 0.6mm and most if not all of your line twist problems should vanish as well.
If it's rocky, you may have no choice other than to use the 0.4mm line but you'd be using because the additional abraision that heavier line can withstand gives you an edge, not because you can put any more pressure on a fish.
Some basic tips on spooling up can be seen here
There are two things that I would add to what's in the video about spooling up. Firstly, it's a good idea to leave a good 7mm or so of tag end on your spool knot. This makes it easy to overwrap the tag end and make sure that it lies flat instead of having a short tag sticking up and snagging your line as you try to cast. Secondly, make the first dozen or so wraps on the reel by hand under a lot of pressure before engaging the bale arm and filling the spool under light pressure. Nylon is elastic and doing this ensures that the first few turns really grip the spindle of the spool. There is no worse feling than cranking up the drag on a big fish only to realise that all the line on the spool is rotating and the spool isn't going round so you have no real control at all - that's mainly a braid problem but I've seen it happen with nylon as well.