How much time do you have to spare in Bishkek? Do you have to daytrip or can you overnight?

If you can do an overnight, I'd suggest trying to make it to lake Song Kul, rather than to Issyk Kul. Issyk Kul is interesting in a sociological way, to see all the drunken Kazakhs holidaying in the resort towns on its shores, like Cholpon-Ata, but Song Kul is an incredibly beautiful little alpine lake surrounded by yurts and rolling grassland. Apparently it's not difficult to arrange a tour from Bishkek, and the travel time would be reasonable, though it might be a bit chilly still at this time of year. We drove through the region last year, take a look here if you're interested. Kyrgyzstan is stunningly beautiful once you get up into the hills, so it would be a great shame to be stuck in Bishkek the whole time, which from what I hear is pleasant but unremarkable (we bypassed it).

If you have the chance to detour to Samarkand you should definitely do it, the Registan is amazing. The fast train from Tashkent sounds like a great option. Bukhara is a beautiful and much smaller place, but you'd have to fly to get there quickly from Tashkent. (Khiva is lovely too, arguably the nicest of the lot, though even further away - and beware the road from Bukhara to Khiva has some of the most diabolically cratered asphalt ever to be called a 'road'.)

If you have even a spare few hours in Almaty it's easy to get up into the mountains which overlook the city.

It's a really interesting part of the world - enjoy!