How important is it to get clipless pedals?

Wade,
 They are very important, and once you use them you'll likey never go back.
Frank.

Get them.  I have yet to ever talk to a cyclist in 20 years of riding that does NOT think clipless pedals are the only way to go.

I think I would avoid the dual use, honestly, unless you know you will be riding in normal shoes a lot.  I think the only few times that clipless is NOT the way to go is BMX riding or as a bike messanger (since they need to get on and off the bike a lot and walk into buildings.)

I have yet to make the jump, but I know that it will be my next gear purchase.. I was hesitant for a long time because I was nervous about being able to get out of them. I finally got over that this summer, but was too broke and too close to race day to change anything major like that.

I know I'm wasting a lot of energy; when my feet slide around on the pedals, that's body generated motion that could be pushing the pedals. My feet are only translating body motion to the pedals during 40% of the stroke (if that); the rest of the stroke, my feet are just riding the pedal to the top so I can push down again. Clips will make for much greater efficiency and hopefully make me faster (nothing could make me slower).

Second, as someone who is rather duckfooted, if I don't constantly think about it, my heels (particularly my left heel) bump the crank arm each time I push down. This is more wasted energy, in addition to being distracting and annoying.

By wasting less energy, you will be able to go faster and longer, which will definitely improve the quality of your workouts.

Before I went platform/clipless, I'd look into Mtn. Biking pedals/shoes If you absolutely MUST walk in your bike shoes.

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I would say... Vital! Probably the the single biggest gain in speed and performance that you can buy. As far as your workout, it may be a two edged thing. Clipless pedals let you use more muscle groups to power your bike so you build/strengthen more muscle with your training, but you are more effecient so your training rides and commute may use a tiny bit less energy. If I had a separate beater/commuter I would use mtb pedals and shoes. I love the crank bros smarty which you could ride wearing sandals or any shoe and have the recessed cleat on a mtb shoe for training during commutes and errands.

Go for the clipless pedals.  After you complete the mandatory falling over once, you won't look back.  I have a set of combo pedals that I used to use on my mtn bike, because I am a big coward on steep trails.  I could see starting with them to get comfortable - keep one foot clipped in and the other on the platform side for a few rides.  But then I would switch to a mtn bike shoe/pedal if I was going to be doing a lot of walking.  When I was on my 10 year hiatus from racing, I switched my bike from Looks to SPD for touring, so I could wear walkable shoes.

Definately go clipless.
I come from MTB and I only use the shimano spd style as the cleat is completely inside the shoe tread so it's fine for walking.

The stiffness of a bike shoe is a big part of the efficiency gain, it's stopping your foot bending over the pedal or looking at it the other way round, your running shoes are allowing your foot to bend over the pedal an your foot is tiring/wasting energy resisting the bending motion on every stroke.

Don't worry about the unclipping, it's a very natural action, if you use/have used toe clips, clipless is much easier.