From what I can see you have four points that need addressing:
(1) "With this in mind, I am not sure I need full enterprise level architecture"
Lets deal with the high level fluff first. It depends on what you mean by "full enterprise level architecture", but the short answer is "Yes" you need to address many aspects of the system (and it will depend on the context of the system as to what the main ones are). If nothing else, the keys ones would be Change and Supportability. You need to structure the application in a way that supports changes in the future (logical and physical separation of concerns (Dependency Injection is a great for the latter); modular design, etc).
(2) "How to properly separate concerns in my architecture without designing a spacecraft?"
I like this approach (it's an article I wrote that distilled everything I had learnt up to that point) - but here's the gist:

Looking at this you'll have a minimum of six assemblies - and that's not huge. If you can break your system down (separate concerns) into these large buckets it should go a long way to giving what you need.
(3) Detail
Separating concerns into different layers and classes is great but you need to go further than that if you want to be able to effectively deal with change. Dependency Inversion (DI) is a key tool to use here. When I learnt DI it was a hand-rolled affair (as shown in the previous link) but there are lots of frameworks (etc) for it now. If you're new to DI (and you work in .Net) the article will step you through the basics.
(4) How to move forward
Get a simple vertical slice (UI all the way to the DB) working using DI, etc. As you do this you'll also be building the bones of the framework (sub-systems and major plumbing) that your system will use.
Having got that working start on a second slice; it's at this point that you should uncover any places where you're inadvertently not reusing things you should be - this is the time to change those before you build slices 3,4 and 5 - before there's too much rework.
Updates for Comments:
Do you you think I should completely
drop web forms and take up MVC from
scratch or just with what I know for
now?
I have no idea, but for the answer to be 'yes' you'd need to be able to answer these following questions with 'yes':
- We have the required skills and experience to use and support MVC.
- We have time to make the change (there is clear benefit in making this change).
- We know MVC is better suited for our needs.
- Making this change does not put successful delivery at risk.
...do I need to move to projects and
setup each of these layers as a
separate project?
Yes. Projects map 1-to-1 with assemblies, so get the benefits of loose-coupling you'll definitely want to separate things that way, and be careful how you set references.
when you refer to POCOs, are you meaning just DTOs or rich domain objects?
DTO not Rich Domain Object. BUT, people seem yo use the terms POCO and DTO interchangeably when strictly speaking they aren't - if you're from the Martin Fowler school of thought. In his view a DTO would be a bunch of POCO's (or other objects(?)) parcelled together for sending "across the wire" so that you only make one call to some external system and not lots of calls.
Everyone says I should not expose my
data structures to my UI, but I say
why not?
Managing dependencies. What you don't want is for you UI to reference the physical data structure because as soon as that changes (and it will) you'll be (to use the technical term) screwed. This is the whole point of layering. What you want to do is have the UI depend on abstractions - not implementations. In the 5-Layer Architecture the POCOs are safe to use for that because they are an abstract / logical definition of 'some thing' (a business concept) and so they should only change if there is a business reason - so in that sense they are fairly stable and safer to depend on.