Both planks and push-ups rely on core strength. If you can't do a good plank, chances are you'll have problems doing push-ups on the toes. I've found the only way to get better is to keep on trying, and push yourself a bit beyond where you think you can go. On planks, try the bent arm versions, and when you feel like you can't hold on anymore, try to get out a few more seconds by pulling in your abs a bit more and bracing the core.
I remember back when I started doing planks (with Aaron Lankford's "PowerKicks" workout). I couldn't hold them as long as he did the entire time (for either of the two sets). Last time I got this workout out, the planks were easy: I held both sets without the break in between, and could go longer. What it took was just trying and doing them, and going a bit longer each time.
If you want to improve in these two areas, you might want to focus on them for a while. For planks, set a certain amount of time that you want to work up to (1 minute) set a timer, and get into plank. Drop down to your knees for a break when you need to (but, again, try to hold out!), then go back up again and repeat until the end of the time limit. Work on holding each plank as long as you can, and shortening the break times. Eventually, you can work up to holding the entire amount of time with no breaks. Then it's time to up the amount of time you aim for!
For push-ups, you can build up to them by doing "negative" push-ups. Lower yourself in the "on the toes" position, then go to knees, and push up. Another technique is to try an intermediate position between "on the knees" and "on the toes." Get on your knees, and lift one leg off the ground, so you are on one knee. I like to use these when I can't do anymore on my toes, but going to the knees seems too easy.
I found that yoga (especially doing the "chaturanga" push-up moves) helped me go from knees to toes, as did trying all the push-ups in P90X. Now, I do most of my push-ups on my toes. (though if I follow Cathe's speed in GS upper body with the very short breaks, I can't, so I make the breaks longer so I can do them all on my toes.)