using words "suggest" and "insist"

Online Oxford Collocation Dictionary suggests using these prepositions with suggest:

PREP. as She suggested John as chairman. | for Who would you suggest for the job? | to He suggested to the committee that they should delay making a decision.

And these prepositions with insist (in fact, these are two meaning basically the same):

PREP. on/upon He insists on speaking to you personally

Also Cambridge Dictionary points out some typical errors of using suggest:

Typical errors

• We don’t use suggest + indirect object + to-infinitive when we suggest an action to someone:

He suggested that I should apply for a job in a bank or insurance company.

Not: He suggested me to apply …

• We don’t use suggest + to-infinitive:

Henry wanted to tell everyone, but Dora suggested waiting until the news had been announced officially.

Not: Dora suggested to wait …

As a result, two of your examples - I suggest to land here and I insist to land here - are grammatically incorrect, while others are perfectly fine.

As for different forms of usage (all from Cambridge Dictionary), with suggest there are quite a lot of options:

• I suggested an Italian restaurant near the station for the party.

• [formal] Might I suggest a white wine with your salmon, sir?

• [ + (that) ] I suggest (that) we wait a while before we make any firm decisions.

• Liz suggested (that) I try the shop on Mill Road.

• [ + -ing verb ] I suggested putting the matter to the committee.

• [ + question word ] Can you suggest where I could buy a dozen roses?

With insist, however, there are not so much:

[ + (that) ] Greg still insists (that) he did nothing wrong.

Please go first - I insist!

She insisted on seeing her lawyer.

P.S. Also please don't confuse these words, they don't share the same meaning! If you are suggesting something, you are just offering one of the possible solutions, whereas if you are insisting, you are saying something firmly or demanding for something forcefully (suggest and insist). If you already know that, it's absolutely great!

To suggest is transitive and requires a direct object.

He suggested beer.

He suggested that we should buy/drink beer. (see https://en.wikipedia.org/wiki/Content_clause and the link to Language Log)

Thus of your examples, only

I suggest {(that) we (should) land here.} = content (noun) clause as direct object

I suggest {landing here.} = gerund phrase as direct object.

are correct.

To insist is intransitive with a prepositional phrase (usually {"on/upon + substantive}) as a complement.

Thus “I insist {on/upon landing here.}” is correct as the complement is a prepositional gerundive clause, whereas

*“I insist {to land here.}” is wrong as {to land here} would provide a direct object (infinitive phrase) and that would be invalid.

But to insist also may accept a content clause as an object

I insist {(that) we (should) land here.} = content (noun) clause as complement.