Slow boot up

It is really not necessary to shutdown your Mac Mini unless you are not going to use it for several days.

Personally I sleep my Mac's 24-7. I only restart them for updates that require it and only shutdown when away for more than a couple of days.

There are several reasons that can contribute to longer restart and startup times.

1. Your HDD is getting full.

2. Added RAM which takes longer to check.

3. Startup Items in your Account and Mavericks resume feature.

How long?

Yes, I think we've all gone through this. I would recommend that run a utilities programme like Cocktail which hopefully will sort out your issues. You could try running Disk Utilities first and check that you reset you browers, Google/Safari which may also help. I confirm that it's not necessary to switch off your Mac Mini. It uses so little power it frankly isn't worth it. I don't believe this to be a software issue with Mavericks at all. If anything Mavericks is quicker than Mountain Lion. You might also think about your RAM, 8Gbs will certainly help thinsg along and it's not that expensive.