You don't need to replace the tire or the innertube. You can patch the innertube, and patching is a useful skill to master, although I usually wait until I have a few tubes to patch, and repair them in bulk. Don't bother with "glueless" patches, and don't be stingy with the vulcanizing agent. Do be sure to clean the area you're going to patch (there's mold-release compound on the tube that prevents a good bond).
Small penetrations as you see here don't damage the tire enough to warrant replacement. If you want to get fancy, you can clean up the tread cuts and squirt a tiny amount of cyanoacrylate glue (crazy glue) into them to repair the tread. If the tire gets a long slash (or other larger damage), such that the innertube could squeeze through, then insert a boot as an emergency measure and replace the tire as soon as you can.
I agree that these Vittoria Zaffiros are not good tires (Zaffiro Pros are actually pretty good). They're also not long-wearing. You might want to budget for replacements. But you don't need to replace them now on account of this damage.
The diameter of your tire is 700C (in conventional nomenclature) or 622 mm (in the ETRTO system). There's no wiggle room on diameter. You can fit a range of widths, and the width of tire you can fit will depend on the rim width (here's WTB's rim/tire compatibility chart), and also on the frame and parts that might interfere with the tire. It's possible, for example, that the rim could handle a 32-mm tire but the frame will interfere with anything bigger than 30 mm. You seem to have an earlier version of the Synapse; you should dig up Cannondale's spec sheet for your model year.