Before going to the store, I put my socks on. After my shoes were on, I left the house, grabbing my keys as I locked the door. I went to the car to get in.
The above is grammatically correct, but I think most people would identify clarity problems. Let's look at how we can more effectively communicate the ideas.
Timeline
I think it's true that most readers would prefer to read things in order. For myself, it is jarring to read part of a sentence only to find out that the second part of the sentence happened before the first part.
As an example, rephrasing our first sentence according to timeline makes it much clearer:
I put my socks on before going to the store.
-ing and "as"
When using -ing words to convey simultaneous action*, we have to be attentive to our timeline. For example, "grabbing my keys as I locked the door"—we need the keys to lock the door, so we cannot grab the keys and lock the door at the same time. As usual, direct language is preferred.
Another example:
Putting my socks on, I slipped on my shoes.
The example above is problematic because we do not slip on our shoes while putting our socks on.
When using "as", we are also establishing timeline of one thing happening at the same time that something else happens. For example, "I washed my hands as I closed the cupboard."—those are two things which do not often happen at the same time. It is important to convey things in the proper order.
So, how do we clearly convey activity in the proper order? By ordering our clauses and sentences, minimizing the use of words which affect the timeline.
I put my socks on, then slipped on my shoes. I grabbed the keys, left the house, and locked the door behind me. I went to the car to get in.
While the example remains bland, it is now very clear what happens and when. By watching the timeline, we've significantly improved this example.
To/Could/Would
"I went to the car to get in."—This sentence tells us the intention, but does not tell us that you entered the car. It only says that you went to the car "to" get in.
Similar problems arise with "could" and "would". Take a look at these two examples:
I could see the rainbow.
I would go to the park on Wednesday.
The first asserts that it was possible to see the rainbow, but it does not say that you actually saw the rainbow. The second tells us that you would do something, but were you going to? It's very unclear. Compare these:
I saw the rainbow.
I will go to the park on Wednesday.
It is now abundantly clear that you saw that rainbow and you will be going to the park on Wednesday.
Very often, I see to/could/would used in a way which conveys an action taken, but none of these usages actually conveys the action taken.
Let's look at our main example with "to" clarified:
I put my socks on, then slipped on my shoes. I grabbed the keys, left the house, and locked the door behind me. I hopped in the car.
While this example is not great prose, it is much clearer and conveys activity in a linear, definitive manner. I hope this discussion helps you in improving your writing.
*Words ending in -ing are not always used to convey simultaneous action. For example, "I love jumping" uses "jumping" as a gerund (an -ing word that is a noun). To keep it simple, ask yourself if it is conveying simultaneous action.