The correct option is B go
'When the verb in the main clause is in the future tense, we often use a present tense in the subordinate clause to refer to the future event.
The given sentence consists of two events: "I will stay at my sister's place" and "I ___ to Shimla". The first action 'will stay' is in the future tense. It will take place at the same time as the second action, hence the tense for the latter should be in the simple present tense.
The words "I will go" describe a future tense, but as the sequence of events suggest, the action must be in the present tense. (The second action actually takes place first, after which the future action 'will stay at...' takes place.) Hence the underlined tense form must be changed and option D is incorrect.
Option A: 'Go' describes the simple present tense. Hence A is the correct option.
Option B: 'Goes' is in the third person point of view, whereas the subject "I" is in the first person. Hence B is incorrect.
Option C: 'Am going' is in the present continuous tense, whereas the tense of the clause "when I ___ to Shimla" must be in simple present. Hence C is incorrect.