Although both « envisager » and « prévoir » can be translated into “plan / be planning”, I’d say they differ in one crucial respect:

1 : J’envisage de visiter ma tante le mois prochain.

Here, the speaker is just toying with the idea, planning it in his mind only, and probably not making any actual preparation for it yet. It remains to be seen whether she will actually carry out the plan.

2 : Je prévois de visiter ma tante le mois prochain.

Whereas with the use of « prévoir », the plan sounds more realistic: she not only envisions the possibility of visiting her aunt, but also she may well have already set some part of the preparation for the trip in motion – by booking a train ticket, clearing her schedule for that day etc.

Or, even if she has yet to go beyond comtemplating the idea just in her mind, she is more likely to realise the plan than « envisager » suggests.

The use of « planifier », on the other hand, presupposes that the plan is due to be carried out. The planning is no longer confined to the comtemplating aspect. I’d rather reserve « planifier » for talking about actually organising something – whether it be planning an assassination attempt, the remodeling of a house, an official visit, an extensive vacation etc.

When you say:

Je suis en train d’organiser / de planifier ma visite à ma tante le mois prochain

it definitely means you already have made your decision to visit your aunt. This rules out using envisager or prévoir as suggested in another answer. Both envisager and prévoir are used for an action you have not yet decided to accomplish.

Organiser and planifier both work here. But to me they do not exactly mean the same.

Je suis en train de planifier ma visite à ma tante le mois prochain

means you are trying to fit the action in your general timetable. You might not have decided on the exact date(s) of your visit.

Je suis en train d’organiser ma visite à ma tante le mois prochain.

means you most probably have decided on the date(s) of the visit and are actually attending to the various tasks that need to be done before you can undertake the visit (buying travel tickets, getting a present, looking for someone to mind the dog in your absence, etc.).

Organiser could be replaced by préparer, but préparer does not bring out the idea of things being as systematically thought out.