I am trying to get some back-links for my client, but am unsure what is the right way to set things up with the other party.
What exactly should I be asking other website webmasters to do for my client’s website?
Do I just ask for any link?
Should I ask them to write a “referral” or “endorsement” or “testimonial” and supply a link as well?
What exactly is Google looking for, and what will help my client the most, if that makes any sense?
It’s looking for “organic links”, meaning links that were put there because the other party really thought your content was great and relevant to its readers. In other words, not setup links. 
Yes, you don’t “set up a backlink”, Call it link building or whatever if it makes you feel better, but that’s SPAM
Much better use of time with much less risk of negative backlash if you focus on Content. and Semantic and Responsive mark-up.
If another business we do business with recommends our services, then that isn’t organic?
Yes, I guess the best links are the ones that are developed because people are so in love with us that they just must tell the world via their website.
But if a small auto repair business does work a local business, or partners with a local parts store, or gets business from a local classic car club, then what is the sin to ask those groups if they would put a link to my client’s site IF would recommend his services?
The owner has a list of maybe 6-8 businesses that refer business to him as it is, and so I thought getting them to write a referral or do a testimonial along with a link to his site would be pretty genuine… 
Nah, it’s not a sin as such. But Google is fairly picky about what backlinks it gives any credence to. It’s definitely not a waste of time to do it, but it may not help search rankings. (No-one really knows, and there’s a lot of opinions on it.)
What you are doing is Marketing, which is fine, and worth doing. Whether it’s SEO is another matter.
A little testimonial along with the link seems a good way to go to me.
I have to get money to buy the software, but I hope to start creating audio testimonials - from real people - later this week or next week. Maybe if another business gives us a back-link, I can see if they will do a brief 30 second audio testimonial as well.
I’m sure Google’s spiders won’t care, but it may help to show people that my client is the “real deal”! 
Hm, but what’s the motivation for anyone to listen to it? I hope it won’t be set to autoplay.
If I am ever lucky enough to get people to visit his new website, then as people look around, when the see the section with customer referrals, I would think they would be interested in see what others have to say. The plan is to have a picture and a written recommendation plus a mini sound-bite of maybe 20-30 seconds.
If I saw that I would click to listen so I could hear what real people have to say…
Trying to make a point?
What software, how much and is there a free alternative?
Edit:
As far as back links are concerned, do the relevant pages have links to Facebook, Twitter, LinkedIn, etc?
I forget the name. It’s for Mac. I think $50.
No links to social media as of now.
My plan is to have a picture of a real person, a real recommendation in text, and a link to that person speaking labeled Listen to Mary’s Recommendation
Certainly not what you have above!
lol
I had already assumed as much. Sorry, my sense of humor is a bit warped at times.
Anyway, I think that is a good idea, (could even toss in a small pic) and I think most visitors that are in surroundings that would allow them to would play at least one. I wouldn’t go overboard and have tons of them but a few would be good,
Yes, as mentioned above, a snapshot - not air-brushed - of the customer.
I was thinking maybe 5-6 recommendations.
My hope is others would welcome “normal” looking people with normal sound-bites versus some air-brushed blonde bimbo… 
Why not? It is relatively simple to include links and they can be “Fat Free” and effective.
I believe most Smartphones have a Free recording App’ which should be adequate and also think the customer may object to additional expense unless it was justified.
I think any client that complains about an additional $50 on a project that is quickly approaching over $10,000 in true value needs to be slapped… 
You may be quite right, but I tend to view anything of this nature as probably a set-up, so I wouldn’t bother listening.
The other problem is that, in my experience, ordinary people are very reluctant to get involved in anything which involves having their photo taken or being recorded in any way. Perhaps you already know of a few more-extrovert customers who would be willing to participate, but if not, I’d highly recommend finding some before you go to the expense of purchasing software. (And remember that, unless you provide a transcript, your audio clips will be as irrelevant to search engines as they will be to hearing-impaired site visitors.)