Help me upgrade my beginner system!

Definitely NEED to replace the cartridge. But most importantly you should spend all the rest of the money on new or used better speakers. Power conditioner is a total waste of a plug at your level. 

Very appreciative of all the passionate advice! I'm going to contact my local store and send them my trade-in list and go from there, as well as get their recommendations. At this point, I've not ruled out room treatments, a speaker upgrade, and/or an integrated amp upgrade.

This will probably take a number of weeks to work through but I'll definitely report back on what I got and the results.

There is NOTHING wrong with the DAC in your Peachtree.  It sounds better than you think.  Of course, if you're hell-bent upon spending money, then by all means do so, but you'll be throwing it away.   I run the early model Peachtree DAC-iT.  Your amp will be running the newer iTx version.

I once had a chance to do a serious comparison with the Schiit Yggy and the PS Audio DirectStream.  The consensus that day was that the Schiit was well...  Most there actually preferred the Peachtree over the PS Audio.  I liked them both and felt the PS might be a little better, but only a little.  And certainly not $6000 worth better.

Your Peachtree amp has plenty of muscle.  The rest of your system is really quite good.  Not extreme hi-end, but will work just fine thank-you.  You would have to spend a LOT of money to improve on your turntable.

As for speakers, well the sky's the limit.  If your current speakers are on stands, then a modest floor standing loudspeaker won't take up any more floor space.  What would be a very good match, be a significant upgrade from what you run now, and would be dirt cheap for what you'd be getting are some used SEAS Thor speakers I saw on a Canadian website, or a pair in Arizona.  They would work great with your Peachtree.

At one time I compared the preamp + amp sections of the Peachtree Nova 150. I had this unit for about 3 years. I forgot how I did the separation of the different parts to test, but I know I did do that and came to the following conclusions:

- the preamp section is terrible

- I think I also thought the DAC was terrible

- the amp section was pretty good, though nowhere close to the Peachtree GAN1 that I mentioned above.

I think it laudable that you are interested in working with and supporting your local audio store. You have some great speaker brands to choose from. I agree with grislybutter that bookshelf speakers would be ideal for that room, and I think would offer the biggest bang for the buck. Not sure that you mentioned what electronic brands they offer. I agree that the Peachtree is the weakest link in your system FWIW.