"spent a lot of time to shop" vs. "spent a lot of time shopping"
Solution 1:
As a crude rule of thumb, if you:
- spend something to do something
... then you spend it before you get or achieve the second thing. Here are some examples:
- We spent a lot of money to ensure the highest quality of workmanship.
- It's time to spend money to create jobs.
- There's no hiding from the fact we spent money to get players in.
In these instances, the infinitives are infinitives of purpose. In other words the infinitive tells us the goal of the spending activity.
In contrast if you:
- spend something doing something
... then you spend it as you are doing it. In other words the spending something and the doing something are happening at the same time:
- We spent a long time cleaning up after the party.
- We spent a lot of energy exercising and fretting over our physical condition.
Notice that in these instances, the -ing-clause tells us how we spent something, not necessarily why we wanted to spend it. Consider the following examples:
- We spent a lot of time getting nowhere
- We spent a lot of time being chased be photographers
Here we did not spend the time with the goal of not making progress or with the goal of being chased.
Gerund participle forms of verbs often indicate simultaneity, in other words they show that two actions are happening at the same time:
- He was run over crossing the road.
- Speaking with his mouth full, he asked me what I'd been doing.
The Original Poster's Question
*We spent a lot of time to shop.
We spent a lot of time shopping.
The Original Poster needs to use the second sentence here. The reason is that the spending time happened concurrently with the shopping. The two activities happened together.
Solution 2:
I agree with Ann O'Rack at WordReference:
It doesn't sound idiomatic to say "I spent 10 minutes to eat my meal"[–] it's definitely "I spent 10 minutes eating my meal".
Even the 'in order to' (ie 'in preparing for') reading doesn't sound idiomatic to me. This would correspond to an unaugmented 'I spent 10 minutes.'.
These Google Ngrams where 'spend some time shopping' manifests but 'spend some time to shop' doesn't, seem to indicate that my gut reaction is in agreement with many other people's.