I would change little. Maybe the 14th to apply to women and sexual orientation and anyone else who aren't treated fairly. Even then, we've added numerous layers of civil rights laws along those lines, so it would be mostly symbolic.
Maybe update the 4th to explicitly disallow mass surveillance of the population through electronic means. We already know the government is violating the 4th.
I'd be good with the gold standard and auditing the fed. Balanced budget, term limits, and line item veto are fair game if we're going to change the rules.
Speaking of the 14th... Corporations aren't people - they are made up of people. They have to be Persons for certain purposes of the law; it is one of the most misreported and misunderstood concepts. A corporation is a Person, for example, if it is minority owned and discriminated against for government contracts. I think we all would agree they shouldn't be discriminated against. (that is the famous case where the Court said "corporations are Persons" ... for the purpose of this ruling). A corporation is a Person in the sense you can sue one. A corporation is a Person in the sense that its papers are free from search without warrants. Do you really want to do away with these (and other equally important) things? When the courts have ruled about corporate Personhood, it's always been very narrow rulings about specific issues like those I mentioned.
If you believe in the 1st amendment, you cannot believe in campaign finance laws. When you have the government choosing who can speak or who can buy a megaphone (through public financing or laws restricting advertising), the ruling party will get to decide. We don't want any part of it. There will always be a loophole, anyway - like if I want to run 24/7 campaign ads, all I have to do is buy a newspaper or TV station. The best we can do is require the speaker to be identified, so we all can take that into account when forming our opinions.
We're not a Democracy. The electoral college is there for a reason, and good reason.