Have you considered using BioPython. They have a sequence reader that can read fasta files. And if you are interested in coding one yourself, you can take a look at BioPython’s code.
Edit: Code added
def read_fasta(fp):
name, seq = None, []
for line in fp:
line = line.rstrip()
if line.startswith(">"):
if name: yield (name, ''.join(seq))
name, seq = line, []
else:
seq.append(line)
if name: yield (name, ''.join(seq))
with open('f.fasta') as fp:
for name, seq in read_fasta(fp):
print(name, seq)