Leading questions:
Representative vs Direct Democracy?
Unitary or Federal?
Presidential or Parliamentary?
How much separations of powers should there be? In presidential systems, such as the United States of America, there is often deadlock between the executive and legislature. In parliamentary systems, the head of government is elected by legislature, therefore, there is practically no deadlock as long as theres is majority support of the executive in the legislature (although, there can still be courts to determine constitutionality of policiss). Would you prefer more checks and balances, but can also result in more deadlock, or a government more easily able to enact policies, for better or for worse?
Electoral method? FPTP? Two-Round? Ranked-Choice/Single-Transferable Vote? What about legislature? Should there be local districts? Single or Multi member districts? Proportional-representation based on votes for a party? If so, how should the party-lists be determined?
Should anti-democratic parties be banned? Or should all parties be allowed to compete in elections, regardless of ideology? In Germany, they practice what's called "Defensive Democracy" which bans any political parties (and their successors) that are anti-democratic. Some of banned political parties include the nazi party.
How easy or difficult should the constitution br allowed to be changed? Majority support or some type of supermajority support?
Should we really elect officials, or randomly select them via sortition?
These are just some topics to think about, you don't have to answer all of them.
Edit: Clarified some things
Wow huge question! I mean for me in the US I think our democratic republic system is great, that is democracy on a local level, and republic on a state and federal level. We're not in Athens anymore! I would love to see our country do away with the filibuster, the electoral college, and to impose very strict laws around lobbying. I think all elections should be publicly funded, which is such an easy way to level the playing field and negate the influence of corporate cash.
Those voting methods you mentioned of course is up to the states to decide already which is how it should be I think.
No I don't think parties should be banned and I wish we could have room for more parties. I admit I don't know why US has essentially only two where other countries have several. I don't know what the hold-up is or the mechanism for change, but I think competition is always a good thing (capitalism, baby!)
Other comments have mentioned ranked choice voting, proportional representation and single transferable vote - these are all voting systems which encourage having more than two parties. The reason we don't have them in the u.s. now is because people know they're throwing their vote away or even helping the candidate they don't like by voting third party.