What partnership recommendations would you give startup CTOs?

It depends on your startup itself. Does it require partnership or not. As a startup, you are mostly dealing with change or behaviour management for customers OR managing cashflow. So partners that can address or help in these 2 areas would be very helpful. 

Also don't be afraid to partner with a "competitor" in the same space as you both are changing the market, and market increase benefits you both.

We identify our long term & short term project objectives before we start looking for partner(s).

For all the limited work like webpages, animation etc., we estimate the cost of the job, and post it on sites that allows people to bid for the job. This helps us keep the cost down.

The rest of the work that's going to be continuous development, we give small module to different partners, and then take a call based on our needs & objectives if we've found the partner we would like to associate long term.

The recommendation is to have partners engaged in the initiatives for sure. Depending on where in the trajectory the tech is, the need would be varying. We might need reasonably more in the initial stages and tapper down as things get stabilized or the team gets built or they have acquired the right skills. Also, this would be a factor on what is the varying part of the budget, it is better to lean on partners for this as we can always add/reduce as the budgets get evolved. 
The recommendation is, to look at the task in hand, how long, depth, and so on, basis that looking for someone who has been there and done that. If someone starts experimenting with you, then there would some level of uncertainties that you will be buying into, if that is fine and it is a factor or other variables (for ex cost, time to acquire people) you might be fine taking the risk.
In short, it is advisable to have a partner who can help bring in the missing capabilities, help with the expedition of the programs, and for sure when the budgets are varying.

I would make sure that they question you hard on the objectives that you set to achieve. Prepare a high-level brief, not a very specific one. You no-doubt will have the detail, but a great way to gauge the potential partners' capabilities to 'walk in your shoes'. The level of empathy to you challenges you detect during these initial discussions are a strong leading indicator for future success. Also explore commercial constructs where their success in making you successful is financially beneficial to the partner. 

Due diligence is key. Listen to, but do not blindly trust, marketing and sales speeches. Test everything, involve in-house specialists, talk with former and actual customers of future partners and providers, work with your legal department for every NDA, licensing and contract you may be involved in.