First part was very straight-forward: read the input in and simulate the navigation. Taking advantage of itertools.cycle made cycling through the instructions very easy :]

Network = dict[str, dict[str, str]]

def read_instructions(stream=sys.stdin) -> Iterator[str]:
    return itertools.cycle(stream.readline().strip())

def read_network(stream=sys.stdin) -> Network:
    network = defaultdict(dict)

    for line in map(str.strip, stream):
        if not line:
            continue

        source, target_l, target_r = re.findall('[A-Z]{3}', line)
        network[source] = {
            'L': target_l,
            'R': target_r,
        }

    return network

def navigate(instructions: Iterator[str], network: Network) -> int:
    count  = 0
    source = 'AAA'
    target = 'ZZZ'

    while source != target:
        source = network[source][next(instructions)]
        count += 1

    return count

def main(stream=sys.stdin) -> None:
    instructions = read_instructions(stream)
    network      = read_network(stream)
    print(navigate(instructions, network))

The second part was also straight-forward: locate the ghosts, and then navigate from each ghost until you hit an element that ends with a 'Z'. The trick to avoid brute-forcing is to realize that the ghosts will cycle (ie. repeat the same paths) if they all don't land on an element that ends with a 'Z' at the same time. To solve the problem, you just ened to calculate the steps for each ghost to reach an endpoint and then compute the lowest common multiple of those counts. Fortunately, Python has math.lcm, so not much code was needed!

def navigate(instructions: Iterator[str], network: Network, element: str) -> int:
    count = 0

    while not element.endswith('Z'):
        element = network[element][next(instructions)]
        count += 1

    return count

def locate_ghosts(network: Network) -> list[str]:
    return [element for element in network if element.endswith('A')]

def main(stream=sys.stdin) -> None:
    instructions = read_instructions(stream)
    network      = read_network(stream)
    ghosts       = locate_ghosts(network)
    counts       = [navigate(cycle(instructions), network, ghost) for ghost in ghosts]
    print(math.lcm(*counts))