I’ve got some suggestions and “tiers” of permanence for you.

Power

First, get yourself something like a vertical rack power strip (like one of these). You can probably notch the shelves to incorporate it well, without adding any required space behind them. This alone will go a long way towards cleaning things up.

Video

This is probably the other biggest mess is the video cabling. Here, there’s a few options.

Permanent, upgradeable, but harder to keep neat and probably a bit of a PitA to run cables

Use some cable passthrough wallplates like these. You’d install one close to your shelving and one close to your TV, then fish cables through. Given the distance, there’s probably at least one stud in the way that would make it a bit of a pain.

**Permanent, cleaner look, probably easier to run and more expensive **

Use HDMI/coax/RCA jack plates and pull cable through attic or basement. May need active cables to avoid issues (differential pair signaling used by HDMI can get finicky).

Semi-permanent

Purchase or make cable raceways. There are some commercial products that replace baseboard or crown moulding. This is probably the easiest route for clean appearance. You can use the risers in the shelf as anchor points to run up or down to your raceway, if you are ok with visibility there. Otherwise, notch the shelves, like the above suggestion for power, and run raceway/square conduit up or down, with ports for each shelf tier.

Less permanent, more expensive

Get an A/V receiver/mux box that you can use as a central connection appliance for the shelves. This way, everything connects to it and you have the minimum number of cables going from it to your TV. I honestly don’t know how much these things currently cost but they used to be pricey on account of being marketed to the “audiophile” segment.

Networking

Try to concentrate as many of your network-capable systems on adjacent shelves as possible. Install a keystone jackplate and either run Cat6 for each device or use a small edge switch and as short of patch cables as you can manage.