Yo, congrats on the S5II! That camera is a total beast, honestly you picked a winner for your first real setup. Chicago is gonna look amazing through that viewfinder... I remember being super overwhelmed by the L-mount stuff too when I first started out but its basically just the name of the hole on the front where the lens goes lol. Since youre on a budget but want that blurry background vibe, here are my top picks that wont kill your wallet:

• Panasonic Lumix S 50mm f/1.8. This is probably the best bang for your buck. Its super lightweight which is great for walking around the city and the f/1.8 aperture will give you that pretty blur for your dog pics. Its usually way under your $600 limit too!
• Sigma 35mm f/2 DG DN Contemporary. I love this one because its made of metal and feels fancy but is still tiny. 35mm is slightly wider so its better for buildings and street stuff than the 50mm. It fits right in that $600 range.
• Panasonic Lumix S f/1.8 Series 35mm. If you want to stick with the same brand as your camera, this is another fantastic everyday lens that focuses super fast. Tbh, I would look at the 50mm first. Its a classic first prime for a reason. You can usually find it on sale for like $350ish which leaves you money for a nice strap or a bag. Just make sure whatever you buy says L-mount on the box and you are golden. Have fun on your trip!

Late to the party but honestly super satisfied with these:

• Sigma 24mm f2 DG DN Contemporary: solid build but kinda heavy.
• Panasonic Lumix S 85mm f1.8: affordable, insane dog blur.

To add to the point above: avoid the Sigma 45mm f/2.8 DG DN Contemporary if you want that blurry background. I was unfortunately disappointed with how busy the bokeh looked when shooting my dog. For street stuff and portraits, grab the Sigma 65mm f/2 DG DN Contemporary instead. That focal length is perfect for detail and that f/2 aperture actually gives you the depth of field you're looking for without being too bulky.

Stumbled upon this today and honestly I feel your pain. Even after shooting with this system for years, I still get choice paralysis every time I look at the new L-mount releases. Like someone mentioned, the naming schemes are a total headache... I was actually in the same boat last week trying to pick a gift for my brother and ended up just as confused as when I started. It never really gets easier to choose. If you want my two cents from a reliability standpoint, just look at the Sigma Contemporary lineup. Over the years Ive found they just dont quit on you, even when you're out in the elements. Or honestly, just grab any of the Lumix primes. They all have the same weight and build, so you really cant go wrong with whatever brand you pick as long as you stay in those lanes. They are built for the long haul. TL;DR: Stick to the Sigma Contemporary series for the best build and reliability in your budget.