I can highly recommend Kandel's "Principles of neural science". I have not had any biology courses since highschool, but I was able to understand it well. It is really a book meant to teach. The style is beginner friendly, and not tedious/boring at all.
It is an expensive buy and a very long read (I think the paper version is close to 2000 pages), but given the complexity of the subject matter, I find it reasonable.
If you decide to go for it (or for any other neurobiology work, actually) get the newest edition. Getting an older one is a false economy here - our understanding of how this stuff works gets better by the year, and there is lots of missing (or sometimes even not-valid-any-longer) information in the older editions.
A few examples of what might make the book hard to understand for beginners
"Principals of neural science" does not require much preexisting biology knowledge. It is sufficient to roughly know that a cell has a nucleus, plasma with organelles swimming inside, and a membrane. But you have to be comfortable with a bit of biochemistry. This is an example passage concerning the behavior of ions in solution. I found this perfectly understandable with my prior knowledge, I don't know if you need brushing up on chemistry or not.
The cell membrane is selectively permeable to K+ because the otherwise impermeable membrane contains proteins that form pores called ion channels. The channels that are active when the cell is at rest are highly permeable to K+ but considerably less permeable to Na+. The K+ ions tend to leak out of these open channels, down the ion's concentration gradient. As K+ ions exit the cell, they leave behind a cloud of unneutralized negative charge on the inner surface of the membrane, so that the net charge inside the membrane is more negative than that outside.
And then there are examples like this one:
Isoprenylation is another post-translational modification important for anchoring proteins to the cytosolic side of membranes. It occurs shortly after the synthesis of the protein is completed and involves a series of enzymatic steps that result in thioacylation by one of the two long-chain hydrophobic polyisoprenyl moieties (farnesyl, with 15 carbons, or geranyl-geranyl, with 20) of the dulfhydril group of a cysteine at the C-terminus of proteins.
Here, my own chemistry is bad enough that I could only understand that a protein is modified by attaching some molecular structure to it, and this makes it possible to attach the protein to the cell membrane. The lack of understanding of the details (what is a geranyl-geranyl?) did not stop me from understanding the rest of the section, so I don't mind, but I know that there are readers who'd have a problem with that. Sadly, I don't know of a source which does not require such knowledge, and if you feel the need to know every single word, you might need to use a reference while reading, or start with a course in biochemistry first before switching to the neurobiology.
Finally, if you don't have a life science background, there might be words outside of the chemistry you might have to look up, or understand from context, but they will be unfamiliar at first. For example, I was surprised to see the word "process" used in the meaning of a physical protrusion from the cell body. But this is not such a high hurdle, you have to either read ahead and puzzle such words out, or use a dictionary. They are not so frequent to prevent comfortable reading.