I would like advice on a small, inexpensive audio

Try this:
Denon DM38S Micro CD System

You can always update the speakers at a later date.

Squeezebox Radio for less than $200. Start with free Internet Radio and Pandora which offer a lot of source choices in all genres and add other online music services if desired. Add a music server on your home computer later. Add active powered speakers via the headphone jack later as well for a sound upgrade if needed.

I use one in my bedroom in place of a traditional clock radio. I can access Internet music sources including most local radio stations + any music on my music server computer the same as I would with Squeezebox Touches on each of my larger audio systems. Good sound out of the box with built in speaker and very good with earphones/earbuds.

My wife wanted a tube amp for the office, and she wanted small. The MiniWatt S1 is about 5" wide, 5" deep, and 4" high, and is built like a tank. Don't think I could have gotten any smaller. I upgraded the stock Chinese valves for NOS Russian power tubes and NOS Mullard amp tubes. I bought the MiniWatt used on Audiogon for $165, the valves on Ebay for a total of maybe $25 or $30...We stream Internet radio through an Aune DAC, picked up used on Audiogon for $100...We still listen to CD's too, listened to through an old (still unmodified) PS1, model 1001, got cheap, and upgraded the power cord for a PA Audio Power Punch C7, picked up on Ebay for $27...Speakers are nice Onkyo's that had been separated from a bookshelf system (model D-N3XA), found at a Goodwill for ten bucks...Interconnects are Morrow MA1, two sets bought on Audiogon for $60, I think. Speaker cables are magnet wire, spool of 21ga. and spool of 19ga. from TechFixx.com (discovered on Ebay) for around $21 total...All in all, a fine little system with all the detail, nuance, and soundstage you could want. It's a terrific rig that I enjoy listening to almost as much as the main system, if for different reasons...
The Cambridge set should be fine, maybe a nice match for something like the MiniWatt, but I would skip the Sherwood--from what I understand, reliability is an issue, as well as sound quality, only fueling your frustration. Still, it all comes in under your budget, as will a similar rig...
If you are patient and keep your eyes open, you will find nice gear at a nice price. Small upgrades--like tubes and cables--can enhance your sound, but only if you begin with quality components. And simple DIY projects, like the magnet wire speaker cables, can lead to more complex projects down the road, when you have saved up a little time and money. I will be building a set of speakers, for example, with full-range drivers for the MiniWatt (the flea-amps really sing with the single drivers). Of course, DIY can save money, but also allow for that touch of pride at the end--I built that...
Hope this helps with your direction. And there is a wealth of information on Audiogon that our beloved members are happy to share--pick their brains on more specific elements of the rig you would like to build and you will end up with more great information than you know what to do with...If you have any other questions, don't hesitate (not that I am nearly as savvy as many people here, just sharing a small-system-building experience), and I hope that your quest goes well; there is enjoyment in discovery. Cheers...