Well, there are a few ways you can do this. You can purchase something from Bose second-hand (it should be cheaper than Amazon for $265) or something that uses the Wave III's AUX or BoseLink input.

• From Bose (Bluetooth Only):
  • Wave Bluetooth music adapter
    • Good: Uses the BoseLink port to power this accessory and to allow the Wave system to control it. Connecting Bluetooth might turn on the wave system (I have not verified this, but it would make sense).
    • Negative: Doesn't work with all Wave systems, but will for the Wave III
  • Wave III dock with BoseLink + Bluetooth Dock
    • Good: Uses the BoseLink port to power this accessory and to allow the Wave system to control it. Connecting Bluetooth might turn on the wave system (I have not verified this, but it would make sense).
    • Negative: Doesn't work with all Wave systems, but will for the Wave III. Requires two items, a 30 pin dock & an item docked into it to work.
  • Bose Bluetooth Audio adapter
    • Good: Uses the AUX input on the wave.
    • Negative: Requires a separate power source to work. Connecting Bluetooth will not turn on the wave system.
• From Bose (Bluetooth & Wifi):
  • SoundTouch Wireless Link adapter
    • Good: Uses the AUX input on the wave. Has both Internet radio, Airplay 2, Bluetooth, Spotify Connect, NAS support, FLAC support, iTunes support, etc
    • Negative: Requires a separate power source to work. Connecting Bluetooth will not turn on the wave system. SoundTouch has been discontinued.
• Non-Bose (Bluetooth Only):
  • Bluetooth Receiver (multiple versions out there)
    • Good: Uses the AUX input on the wave.
    • Negative: Requires a separate power source to work. Connecting Bluetooth will not turn on the wave system.
• Non-Bose (Wifi):
  • Sonos Port
    • Good: Uses the AUX input on the wave. Has both Internet radio, Airplay 2, Spotify Connect, FLAC support, etc
    • Negative: Requires a separate power source to work. No Bluetooth. Connecting WiFi will not turn on the wave system. Expensive.

Turntables and Speakers

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Turntable: AT-LPW30TK $329 $199.

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Plus powered speakers with audio cable input for the turntable plus Bluetooth for streaming audio.

• New 5.25" Neumi BSP5 $159.

Or better used options:

• 5" Edifier R2000DB $200.
• 5.25" AudioEngine A5+ $245 with stands.
• 4.5" SVS Prime $250.

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Or passive speakers and a used stereo receiver:

• New 5.25" JBL Studio 530 $599 $199 on sale.

Used, compare to buying new speakers for over twice the price:

Towers, then you don't need to buy stands:

• Large 7" Boston $125 https://philadelphia.craigslist.org/ele/d/hatboro-boston-cr95-audiophile-tower/7609505929.html are a good value.
• Dual 5.25" Infinity towers offer $185 https://www.facebook.com/marketplace/item/608623330895678

Or stand mount speakers with included stands are a good value:

• 6.5" Infinity $120 https://philadelphia.craigslist.org/ele/d/reading-infinity-entra-one-bookshelf/7609967037.html
• 5.25" Polk $100 https://philadelphia.craigslist.org/ele/d/hatboro-polk-rti4-speakers-with-stands/7608876192.html

Or more stand mount speakers:

• 5.25" Polk $45 https://www.facebook.com/marketplace/item/1168768734007352
• 6.5" Polk $125 https://philadelphia.craigslist.org/ele/d/philadelphia-polk-rti6/7596421767.html
• 5.25" Wharfedale $180 https://www.facebook.com/marketplace/item/1137967510249780
• 6.5" ELAC Debut $200 https://philadelphia.craigslist.org/ele/d/philadelphia-elac-debut-20-b62/7601113690.html
• 5.25" Q Acoustics $200 https://www.facebook.com/marketplace/item/1589794704852082
• Smaller 4" Mission for a small room $80 https://philadelphia.craigslist.org/ele/d/philadelphia-mission-ms-50/7606968934.html

Plus a stereo receiver to power the passive speakers:

• https://philadelphia.craigslist.org/ele/d/new-hope-sherwood-rx-4109-stero-receiver/7607553273.html with remote
• https://philadelphia.craigslist.org/ele/d/philadelphia-marantz-stereo-receiver/7599230174.html with remote
• https://philadelphia.craigslist.org/ele/d/warrington-denon-stereo-receiver-and/7603393346.html offer $60 plus optional CD/DVD player so offer $80 for both.

Optionally add Bluetooth $22 or better WiFi Streaming with voice or app control + Bluetooth $22 or Airplay 2 with Siri + WiFi + Bluetooth $85.

Or an integrated amp with Bluetooth:

• https://www.facebook.com/marketplace/item/614506377194172

Plus speaker wire, pure copper oxygen free 16 AWG or optional thicker 14 AWG. Then How To Strip Speaker Wire and How to Install Speaker Wire.

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Or a $40 starter setup with small speakers for a small room https://philadelphia.craigslist.org/ele/d/west-chester-small-inexpensive-stereo/7602743790.html optionally add Bluetooth or WiFi or Airplay 2 from above.

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Audio Guides, Speaker Placement, Accessories and Vinyl Care