Oh man, I lived there for two years, and I miss it every day. It’s the best city I’ve ever been in. You’ll have a great time. If you can, get a train pass (Suica or Pasmo, works like an Oyster card) which will negate issues with ticket machines (although they are easy once you know what you’re doing). It’s really easy to get around, but finding specific addresses can be impossible - the building numbering system is, ahem, opaque.
You might be a little early for cherry blossom if you’re leaving mid-April, all depends on the weather. If you’re lucky, good spots include the banks of the river in Nakameguro, and Aoyama Cemetary, which you can turn into a walk up Omotesando, through the Meiju Jingu shrine, back to Harajuku and then on to Shibuya. That’ll take the best part of the day, but you’ll see a load of ace stuff. Go up the top of the Mori Tower in Roppongi for some spectacular views (plus you will be very close to my old office). Other favourite areas were Daikanyama (good for arty stuff), Azabujuban (my old hood), Nezu and Yanaka, which are very old fashioned and the more interesting for it, but almost everywhere has something interesting about it. The Yamanote Line is your friend - there’ll be something worth seeing on almost every stop. I could walk that city for years. It’s so full of contrasts - you can be on a super busy thronged neon overload street, and then just a few corners will take you to a quiet backstreet with a small temple or some lovely little shops. The food is brilliant. Go to Nirvanam in Kamiyacho (near the Tokyo Tower) for the best Indian food I’ve ever had.
There is more than enough to do in the city to keep you busy for months, but as far as getting out of the city goes, I’d say go to Hakone. It’s a volcanic valley an hour or so out of Tokyo, and there is an amazing open air sculpture museum there. Plus you can go transport mad - a typical trip would be a train out of Shinjuku, funicular railway up the mountain, then a cable car over the valley and finally a ride on a replica pirate ship over the lake (I don’t know why it’s a pirate ship). It’s also a good place to get views of the elusive Mt Fuji. Kamakura is easy to get to as well, and has some excellent old temples as well as a giant bronze Buddha (and an eating establishment which advertised itself as an Amish restaurant. I have no idea what they served). Mt Takao is also a good spot, very popular with Tokyoites at the weekend, so maybe go in the working week if you do.
I’ll stop before this becomes a full essay, but I’m sure I’ll think of other stuff. These were just the first things that occurred to me, with the caveat that I haven’t been resident there for a good few years now and some things may well have changed. Just shout if you have anything specific you want to know about and I’ll help as best I can.