don't start with fallout 1 or 2 they are a totally different type of game very old and you won't like em, if you only played NV you're kinda missing what a bethesda fallout game is, i would start with FO4 and spend the next year playing that and then you can always play fallout 3, the bethesda versions of fallout focus heavily on exploration, combat, unique and interesting locations on the map well suited for tactical combat, sneaking around etc, NV you're just in the middle of nowhere all the time, i think you'll enjoy fallout 4 much better than NV.

Not sure how much effect Fallout 3's story has on Fallout 4. There is the Brotherhood of Steel, but we don't know if they follow the ideals of searching for technology no matter the cost from Fallout 1 and Fallout 2's BoS or if they follow the ideal of helping everyone with their technology from Fallout 3. Other than that, there is Harkness from Fallout 3 which introduced the Institute and Synths.

The only reason to play the previous games besides having fun is to catch up on the backstory and it doesn't look like there will be much Fallout 3 backstory in Fallout 4.

Start with one and appreciate what each one added. You can buy the anthology and keep yourself busy for the next few days.