I have older 1988 Miele road bike with Shimano (7 speed) 600 tricolor group set(hubs and all).
I got a pair of Bontrager wheels(Race Xlite)given to me.
I want to go to 9 speed, as will fit on rear hub. I have read plenty that say rear deraileur should work fine, as well as front deraileur. Big question is which STI levers should I look for? 205, ultegra?

Also what brake calipers would be good to upgrade to?(if I am understanding correctly, thus 600 tricolor stuff is old day equivalent to ultegra) just what I've heard.

You can still use your rear derailleur but you will need a 9-speed shifter and chain. Any 9-speed STI will work but since you are upgrading, why not go all the way to 10-speed?

Are your current brakes single or double pivot? If single pivot, any of the newer Shimano double pivot brakes will work well but not the newest (5700, 6800) since they use a somewhat different cable pull. Look for 105 BR-5600 or Ultegra BR-6600 as a suitable set.

I belive they are single pivot.

Will 10 speed fit the same rear hub?(on the Bontrager wheel)
The wheel fit right it the frame without having to spread or cold set it too.

Yes, 9 & 10 speed fit on the same wheels.
I would think you can choose any STI levers you want. Just make sure that levers, chain & cassette are all 10 speed (or all 9)

The pull ratio for 9/10 will work with the old rear derailleur ?

The frame should be cold set.

Tiagra 9-speed stuff works well and is available in new condition. Why go any higher?

Your calipers are likely fine. Remove and clean them, and replace the pads with KoolStop dura salmon cartridge pads. That'd be the most effective braking upgrade.

You'll need a cable stop for the downtube, too.

So, that's shifters, cable stop, chain, cassette, cabling and brake pads. Should be well under $200.

Say again what's wrong with the 7-speed setup? Why not stay with 7-speed and go to the ST-A070 brifters? Cheaper...

Measure your rear frame spread. it must be 130mm to have an 8+ speed rear casette hub slip in.

126 was the last , standard .. when 7 cogs on the rear hub were enough for bike sellers to promote.

AFAIK, any indexed 7-speed Shimano rd will work with newer 8,9 or 10-speed shifters with the sole exception of 8-speed and earlier Dura Ace.

OP: yes, those are single pivot brakes and any newer double pivot will work a lot better. However, what is the "reach" on those brakes since Shimano made the 600-series in both short (37-47 mm) and long (47 - 57 mm) versions? You have to match the reach and short reach double pivots are far more common. Long reach brakes are still available from Shimano and Tektro but are usually special order items.

The OP said twice the frame already takes the new wheel with no problem so there is no need to cold set it.

The calipers are BR 6400, 39min-49max

Got a decent deal on used set of Ultegra 6600 calipers, ordering Ultegra 11-28 cassette(10speed)(new) and Ultegra chain,($52,$22 respectively). Still needing a decent set of shifters though, any heads up would be appreciated

Those Shimano calipers look fantastic and function perfectly well. It's your choice to "upgrade" but make sure they get a nice home.

I had a set of those BR-6400 600 single pivot brakes I got at a very good price and installed on an '85 Bridgestone 400 as an "upgrade" (mostly for looks) over the Dia Compe OEMs. The did work as well as any single pivot brake I ever used but my first experience with double pivot brakes made a believer out of me.

I will admit that I have zero experience with using double pivot brakes, as I started with centerpull, when to sidepull and then to cantilever. But it makes perfect sense they would provide the same power with less effort, as they combine the shorter pivot to pad distance of a centerpull with the direct cable connection between arms of a sidepull. Maybe once I'm retired I'll get a 2nd performance bike in addition to my tourer and I'll find out first-hand.

As has been discussed to death in this forum in the past, any brake that can lock the wheels of a bike is plenty strong enough and decent single pivots can certainly do that. The benefit of double pivots is that you can get the same level of braking with less hand pressure. That's not important in a single stop but can be a real benefit on a long down hill where continuous or frequent braking is needed to moderate your speed. The reduction in hand fatigue is very apparent and very worthwhile.