Clips would be the name for the basket like thingy you screw onto the pedal to position a strap that can be tightened to secure your foot to the pedal. While still available, what most people (by far) use these days are pedals and shoes with cleats instead. As clips don't really need bicycle-specific shoes I assume you're talking about a SPD-style shoe retention system.
What to get differs greatly. As you've already determined, MTB-style offers the advantage of the possibility to walk reasonably normally at the cost of maybe getting the force concentrated on a smaller portion of the foot. Although that has never troubled me.
For low cost, I'd go with Wellgo. I've had both their shoes and their pedals. They broke eventually, but not before providing their money's worth.
My current favourites are Crank Brothers. Their Egg Beater model can feel a bit unstable at first due to the small support surface, but that's easy to adapt to. And there are other versions that offers a bit of a platform should you want one.
While usually not that important on a road bike, the CB mechanism is close to impossible to jam up, which is a nice feature for snow/mud riding.
Time pedals are real similar to CB in terms of the mechanism.
Nothing inherently wrong with Shimano either. Big S, being the dominant on the market, will give you the best chance of being able to just get on a random bike for a test ride.
And while I do appreciate riding with SPDs, don't expect the ride to get much easier, that's not really what it's about.
Shoe retention mainly means that you can pull a little on the upwards bound leg so that more of the power from the downward leg goes towards pushing the bike forward as opposed to lifting the rear leg. W/o shoe retention, you'd need to keep some weight on to prevent your foot from slipping off.
Sure, you can use that pull to add some forward motion too. But the ergonomics of that pull is so much worse than the power you can get from pushing down that the effect is quite marginal.
