Why do intelligent design people think that just because the first cell had a lot of information in it...

...it didn't develop from something simpler?

From what I understand, theories of abiogenesis describe how simple, non-living, organic structures slowly developed into what became the first cell by processes of competition. Those structures that were better at replicating survived and continued to do so until the first fully formed cell emerged.

Have I got it completely wrong?

Intelligent design believers have this topic close to their heart. That is why they often create propaganda - Such as the creationist museum to convince others and THEMSELVES in what they believe.

But yes you are correct with your statement on abiogenesis.

Another point that creationists hold to, is that they BELIEVE that a cell simply "fell" together, which is perfectly possible. But most likely, the first universal ancestor was a much similar lifeform.

It's just chemistry.

Your reasoning is essentially correct. However, there was a large amount of chance in the process as well. It was not so much that the chemical compounds were better at replicating, because compounds do not replicate. Eventually, the compounds by chance came together in such a way that a very simple organism (much simpler than a human cell or probably even a modern bacteria) emerged. This organism was apparently able to reproduce and life as we know it eventually emerged.

"by processes of competition." change to - replication with variation.
"Those structures that were better at replicating..." - change the word replicating to adapting

Good question, shows you are thinking, unlike some folks we know.