Newb Q: Swimming

Yes, that will be no problem to swim that distance with proper training.

Your goal will be to finish, not race the swim, so all you need to do is learn a swim style that expends minimal energy for forward propulsion which means you have to learn to float effortlessly face-down flat in the water. 

Once you can do that horizontal float with minimal kicking, you can add in the swim strokes and a small breath, and you'll make the distance very quickly. The key is to NOT fight the water at this stage - you want everything to feel easy - if you're gasping for air, or kicking a lot, or flailing your arms/torso, you're doing something wrong.

The Total Immersion book has good float drills (you can look 'em up on the web as well) that will allow you to readily swim (slowly) for a long time once you get them down.  

Practice as much as you can , get in the pool as often as possible, preferably over 3x/wk. You won't be pushing yourself aerobically or muscularly on these workouts anyway so you can do unlimited amounts of them without fatiguing yourself.

Getting faster is a whole nother story and is a LOT harder....

I got a 4 month Gym Membership in January and started training for a Half Ironman.  I hadn't done any swimming to speak of for about 10 years prior to that and even then had never been on a swim team or done any type of swim training.

My goals when I started were a little different than your.  I wasn't worried about drowning.  I am comfortable in the water having spent many hours in community pools on hot summer days growing up and time swimming in the open water of rivers, lakes, and the ocean when we were were on vacation and even complete the BSA mile swim three times as Scout camps.  I never really was competent in any of proper stokes other than the elementary back stroke and side stroke but I could do breast stroke or front crawl if I kept my head above water.   My goal was more than to just not drown, I wanted to learn to swim proper freestyle and to be an efficient open water swimmer triathlon swimmer.

My first time in the pool didn't go as well as yours.  I could only go about 30 feet of freestyle before I would panic for air and have to stop to catch my breath before trying again.  After than I dedicated two week of pool time to doing breathing drill.  Once I was able to make some headway on the breathing everything else came quickly.  My approach was very similar to jhaack39.  I would watch videos one day and then swim the next.  I would really pay attention to what I was doing in the water and search for drill to help me in the areas that I was struggling the most (i.e. body rotation, parts of the arm stroke, kicking, etc.).  By 10 weeks I had improved my 30 feet of continuous freestyle to 1200 yards and my speed from 4:00/100 yards to 1:55/ 100 yards.  On race day at 16 weeks I completed the 1.2 mile swim at a 1:42 min/100 yard pace achieving my goal of swimming freestyle the whole distance and beating my goal for time by 3 minutes.    

So...don't be discouraged your first day in the pool.  Stick with it.  I made huge improvements with 3 day a week in the water.  

Yes!  You can do this.

Here's some good info: http://ruthkazez.com/swimming.html   She has a plan for 0 to 700 and 0 to 1650 (a mile).

Any in-person lessons would be great, but don't wait to get started on the consistency of swimming regularly.   Just keep at it.  When you have an opportunity for feedback take it, and work it into your next session.

One newbie advice thing:  It's okay to take breaks.  I stop after every 100 when I am just getting back into swimming to get myself together and have a little reset.  I gradually grow the sets, but I never just swim straight through.  I find my swim doesn't degrade as quickly with small breaks, like 10-20 seconds.   You will not miss them on open water day.   But notice when you stop - are you wildly out of breath? If so, you are swimming too hard.  Swimming takes effort, but you will not be able to keep going at a pace when you feel gassed.  Slow it down.  It's okay, you won't sink. 

Good luck!

Some good advice already, but to add my thoughts:

- If you aren't able to work with a coach, then try to get some feedback from another accomplished swimmer.  There's often a big gap between what we see in our mind's eye and what we're actually doing.  Some objective feedback, though tough to hear sometimes, can be extremely useful.  When I was first starting to train for triathlons, I had a friend who was a former collegiate swimmer and masters all-American who would watch me for a few minutes once a month or so and offer feedback and suggestions.

- It takes time to ingrain muscle memory for a good swim stroke.   Work on establishing a smooth, consistent stroke before worrying about pace.  As the saying goes:  "Slow is smooth; smooth is fast:.

- Swimming technique is a perishable skill.  Frequency is the key to reinforce technique,  Most folks need to be swimming at least three times a week to improve.

Good luck,

Mark