Tiger 1 or Tiger (P) are tier VII tanks, which you could probably unlock in about 1-4 months of dedicated grinding, 5-6 if you're casual about it. I'd highly recommend going for the tank you want to drive, you'll have more fun.
The whole German medium/heavy line can brawl or snipe, once you get to tier V and the Pz IV or VK 30.01H. The lights leading to tier V are not the best, but are useful learning experiences. And I'd also recommend you get a Pz I C because that little troll is FUN.
If the tech tree is a bit confusing for you: to get either the Tiger (P) or Tiger I, you'll need to go for the Pz 35 (t) > Pz 38 (t) > Pz 38 nA (which is annoying to grind) > Pz IV (which is fun) > than choose a VK 30.01 P if you want the Tiger (P) or H if you want the Tiger I.
I prefer to upgrade everything in a tank before moving to the next one, but if you want to go straight for the Tigers, look at the research tree for each tank to see what you'll need to upgrade to unlock the next tank.