I know that 1 byte = 8 bits.
Can I access 1 bits in memory or it should be a full byte?
Let's suppose the memory starts at: 0x0000 then what's the next position? 0x0001 or 0x0008?
It depends on how many bit processer you have. I don't think there are any 1-bit processors around.
But a 8-bit processor would read a full byte, and then you can mask bits to actually read the only one you care about.
Therefore, considering the most common architectures, the next address would be 0x0008
Edit: Based on the comments, there are 1-bit processors