Let's say it's early February.
The time of year won't matter too much by 1900; travel is pretty regular and there are a number of other people heading this way, some of whom are publishing travelogues of their voyages during this time. It'll take you about a month to get there, though if the weather is good and you time your trip to have minimal layovers, it could be done in 3 weeks.
This might be more brief than you were looking for, if so let me know and I can add a bit to it, but to be honest the journey from either France or England to Tahiti involves lots of dull monotony sitting on ships and a bit of time on a train watching mostly flat country whiz by.
Step One : Catch a ship to New York, you can do this in Manchester or head down to London or Southhampton, the voyage will take about 5 days to a week depending on the weather. Once you arrive in New York, customs awaits. If your experience is anything like some of the other Brits who wrote about the journey then you can expect that to take the better part of a day as Customs officials open and examine your travel trunks (unless of course you know in advance to tip the customs agents-- there is a reason the US was referred to as the Land of the Dollar). Once out of customs you would need to arrange a carriage service to get your baggage to your hotel.
Step Two: ~3200 miles of rail travel from New York to San Fransisco with a change of trains in Chicago. You probably aren't in Chicago more than half a day; looking through George Griffith's experience in 1900, he was in Chicago about 3 hours and thought that was too much time-- for the rail yard was amongst the great factories of the city which had the appeal of an 'architectural fungus.' In fairness to Chicago and Griffith, he did imagine the rest of the city must be nice, but it was clearly not a place for a long layover. From Chicago you would take the Overland Express to San Fransisco. All and all, the train journey would have taken 4 or 5 days.
Step 3: Hopefully you've timed your trip so that you only have a few days layover in San Francisco, because the ship you'll take, the Mariposa, only departs from San Francisco every 36 days as it steams between San Francisco -- Papeete (Tahiti) -- Aukland and then back again. The voyage to Papeete takes 10-14 days depending on the weather. So by early March, there you are, in Papeete, in the midst of a South Pacific summer.
You could also go the other direction-- heading to South Africa and then Australia/New Zealand and on to Tahiti, but that trip will take about twice as long. However, you would have the advantage of staying within the bounds of the British Empire until you got to Tahiti. So less issues with customs and no money changing, but a lot more time spent on board ships.
Here are a few of the travelogues published around this time--
Hort, Dora. Tahiti: The Garden of the Pacifc. London: T. Fisher Unwin, 1891.
Reeves, Edward. Brown Men and Women or South Sea Islands in 1895 and 1896. London: Swan Sonnenschein & Co., 1898.
Griffith, George Chetwynd. In an Unknown Prison Land. London: Hutchinson & Co., 1901.
Pallander, Edwin. The Log of an Island Wanderer: Notes of Travel in the Eastern Pacific. London: C. Arthur Pearson, Ltd., 1901.
Keeler, Charles. Tahiti the Golden. San Francisco: Oceanic Steamship Company, 1902.
Agostini, Jules. Tahiti. Paris: J. André, 1905.
Senn, Nicholas. Tahiti the Island Paradise. Chicago: W. B. Conkey Company, 1906.