The design of Bitcoin is such that users can just be users.  

For example, if I transfer a coin to you, you can immediately verify my signature.  You do not need to trust me or trust anyone else -- the transaction itself contains cryptographic proof that I transferred ownership of my coin to you.  You can even follow the chain of signatures backwards and confirm that the person who I received the coin from also signed the transfer, and so on and so forth, as far back as you want to go.  You do not need a network node to do this!

The only risk to you is that I may have also signed that same bitcoin over to someone else.  This is the double-spend problem that had not been solved until Satoshi.  Satoshi's revolutionary solution was to use proof-of-work to time stamp transactions into a chain such that -- if his proof-of-work conjecture held --  that the same coin could not be spent twice.

As a user then, the only additional piece of information you need, is whether the transaction I gave to you was accepted into the blockchain.  A network node can provide you this proof (that you can trustlessly verify for yourself!) with a few kilobytes of information.

Here is a talk I gave (only about 10 min long) that explains how SPV works in more detail (sorry about the crappy audio!)  

https://youtu.be/m7cvPvtGIUI?t=459