To find everything reachable from a starting node, keep going to unvisited neighbours, and remember where you've been so you don't go round in circles. Depth-first search (DFS) follows one path as far as it can, then backs up and tries the next. It's natural to write recursively:

graph = {
    "A": ["B", "C"], "B": ["A", "D", "E"], "C": ["A", "F"],
    "D": ["B"], "E": ["B", "F"], "F": ["C", "E"], "G": [],
}

def dfs(graph, node, visited=None):
    if visited is None:
        visited = []
    visited.append(node)
    for nbr in graph[node]:
        if nbr not in visited:
            dfs(graph, nbr, visited)
    return visited

print(dfs(graph, "A"))

The visited check is essential: without it, A visits B, which visits A, which visits B... for ever. (For big graphs, use a set for visited, so checking is O(1).)

G never appears: nothing connects to it.