On PR the issue is that it doesn't really work with our constituency system. We don't elect a Prime Minister we elect MPs and the party with the most MPs forms a government and the leader of the party becomes Prime Minister. It's a complicated system.
Take an example of where there are two constituencies voting. One votes 51% Conservative and 49% Lib Dem. The other votes 51% Labour and 49% Lib Dem. In this case Lib Dems have 49% of the vote and no MP, where are Conservatives and Labour have 25.5% of the vote and 1 MP each. But how can you give the Lib Dems a seat when neither constituency voted for them with a majority, you can't force a constituency to take an MP they didn't want.
For me we should use transferable voting, where you number candidates from one to five etc...and the votes for the lowest candidate is transferred to the next preferred candidate until some has 50.1% of the vote. This would enable tactical voting and allow people can vote for their preferred party and still vote against the party they least want. Now at the moment tactical voting is far too prevalent and we get MPs not because they are the best, but because they are the least worst. It would also allow for better political analysis on where voters preferences actually lie. I would also add Reopen Nominations to the ballot. This way people who are unhappy with all the choices can still protest, but can also still choose the least worst option. (Also it's fun to see how if RoN could actually win in a constituency).
Next I would reform the house of Lords. I think it's been discussed in detail before, but for me it should be at least in part elected and this would be proportional representation. This way it actively reflects the will of the country. I'm not sure about long term independent members or people like the lords, because it then becomes how are they chosen and for how long. Just look at the issues with the American Supreme court. It's meant to be independent and neutral, but is completely political and partisan. How could we ensure independent members of the house of lords are truly that?