What to purchase first amp or headphones?

If I were you, I'll get k701 first, then adding dac and amp to feed it.

Because when you get K701 (even with your current amp), I think it'll give "more" upgrade than you get good amp to pair with your  k171. (though I never K171, so actually mu suggetion isn't really valid)

I have Technics Home Amplifier which is really good when I paired it with K701, I think it's because home amplifier usually have lot of current which is can make k701 happy.

Just the K701 now and use them out of your 'Sony home amplifier' - I'm assuming a it's a multi-channel receiver, which usually isn't the best for headphone use because they usually have afterthought opamp headphone jacks, but it beats nothing. 

get best source first, then phones, then amp, then cables.

Look at it this way:

If you get the cans first you can plonk them on your head and listen. You can enjoy them where they are right now and also be able to note how they improve when you upgrade your source equipment.

If you get the amp first, you can look at the amp, and go 'Gosh that's a nice amp'. You might even get to plug it in, maybe switch it on. Obviously you could listen to other cans through it, but it'll only be a slight upgrade for lower-end models.

Headphones first - as you can always use them anyways while waiting for the amplifier.