By itself, you can say either one. I would say that "Let's get started" means "Let's get underway," whereas "Let's start" means "Let's begin." For many activities, those two phrasings are pretty much interchangeable, but not always.
When specifying what activity will be started, though, whichever option you choose might affect how the rest of the sentence is structured:
- A prepositional phrase often follows "Let's get started...", whereas that's not necessarily the case for "Let's start..."
- An -ing word can follow "Let's start...", whereas the sentence might sound better with that word omitted after "Let's get started..."
- Words can sometimes be added to "Let's start..." but inserted between "Let's get started..."
For example:
Let's start the test.
Let's get started on the test.
NOT: Let's get started the test.
- Mechanic, to other mechanics in the shop:
Let's start working on the car.
Let's get started on the car.
Awkward, but not ungrammatical: Let's get started working on the car.
- Race official, to runners:
Let's start this race!
Let's get this race started!
NOT: Let's get started this race!
Let's start reading before it gets too late.
Let's get started on the book before it gets too late.
Other times, though, the wording won't change!
- Golfer, looking at the ominous clouds
Let's start before it rains.
Let's get started before it rains.
It's tricky; I can see how a non-native speaker might be unsure about this.
Actually, "from the top down" is a perfectly good phrase which probably means exactly what you want. I don't know why you think it's 'not established'; it's probably not as popular as "from the bottom up" but it's widely used and well understood.
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