Making Voting Day a national holiday would be good.
I have a solution to identifying illegal immigrants - a national database of fingerprints/DNA, which does not have personally identifiable information in it.
When a child is born in the US (and as a part of getting its birth certificate), it gets a DNA swab taken and a lab processes it and sends (say) three-dozen markers to the central database. As part of the ceremony where an immigrant is granted citizenship, their DNA and fingerprints are added to that database. When someone enters the country legally on whatever visa, their fingerprints/DNA are taken and they are added to that database with whatever expiration date is on their visa. If someone is found without ID and is a suspected illegal immigrant, just take their DNA and check it against the citizenship database - it comes back with a yes/no, and they can be deported immediately with no further investigation. As an add-on, their fingerprints/ID get added to the "bar entry" database and they're never allowed legal immigration into the country ever again.
Probably too much to go into a constitutional amendment, though.