I know that species "transform" into other species through the process
I don't think you are thinking about that right.
We have a population of organisms that are breeding together. We humans label that a 'species'. That population has descendants, and over time, the population changes, and when we humans think that the new population look sufficiently different from the old, we label them a new species.
Many species around today still co-exist with the species in which
they evolved from, right?
You might have founder situations, where a small sub population breaks off and inhabits a new environment, and changes a great deal more over time than the original population does, such that we continue to label the original population the same species, while we label the offshoot a different species. But I think what happens more often is that the alleles that cause humans to label a population a distinct 'species' dwindle to nothing in the descendants, and then we say that the original species is extinct, replaced by the new one. but that's a matter of labels: there is continuity in that the new 'species' is just the descendants of the older one.
Actually, is it correct to say that all species coexist with some
No. Can you even name any examples where you think this is true?
if the monkeys that we evolved from are still around today,
What on earth makes you think that's remotely true? For one thing we aren't descended from monkeys. We are descended from primates that were more like apes, not monkeys. There is no living population of primates that we would categorize as being the same species as that non-homo sapiens ancestor population.
Are there still early humans, like Neanderthal's or Hominids or what
have you still roaming around some where?
I'm confused how you think an organism or a population existing in the 21st century could be "early". All the hominids walking around today are members of homo sapiens sapiens.