I am a new/recent landlord who had to go through this about 6 months ago in a similar situation.
My first and only rental property had about 20 or so enquiries, and I had posted the rental ad online using zillow's rental manager.
I responded to each one using the rental manager tool which is a bit clunky for messaging, but it kept track of responses automatically. However, it was tricky to manage the candidates that called me without any digital trail. I solved that by emailing them a somewhat canned email (this is a gmail "labs" feature like Shaun mentioned) followup response that also saved me time, and filing it away in a folder I could track.
I did a few one on one showings just because I didn't know any better, and couple group showings (open house) at a time that was convenient to me, to save more time and handed each of person a paper rental application on arrival.
However, I didn't take a fee or make them apply on the spot, like some people suggested me to do.
Rather, what I did was let each applicant know that I'd be using a tenant screening service called Smartmove by Transunion, that they would pay for as the application fee, but I would reimburse them if they ended up being the successful tenants. This had the impact of filtering out only the serious candidates.
Using their email addresses, I was able to use the smart move service to send a screening request to the applicants. Note, that you should get names and email addresses of everyone who is going to be signing the lease so you can run their backgrounds individually.
That cut down the number of qualified applicants a lot, and I was able to do the full credit check and get the best candidates.
One family I considered didn't qualify according to my criteria, but I considered them since they were working with me to produce the deposit. They got me a partial deposit, however, when they couldn't deliver the full deposit when they said they would, I had to be unemotional about it and say no to them. Then, I started looking again, and found the right tenant a week later.
Now 6 months in, I check in about once per month in person, and keep an open communication channel responding ASAP to any questions my tenant might have, but there haven't been many so far. I'm learning every day from this forum how to be the best landlord, and loving it.
Good luck on your journey Andrew. I hope this info about my experience helps you.