My criteria for a new city
Most Important
*** I want to have a mostly local lifestyle that doesn't require frequent car use. I want to bike to work. Having the option of walking or biking to social get togethers and leisure activities is also important. Which brings me to..
*** For leisure, good surfing and kayaking are a must. I'm being picky here, I know. I did enough of both as a kid to realize how much I miss them living in the Midwest.
*** Needs to be a place I can find a steady career. Ideally a large metropolis, no resort towns.
*** Good nightlife for young professionals
Also Important
* Bike friendly
* Diverse
* General quality of life
* Cost of living (I realize this is a tough one since I'm asking for big cities with beaches. As long as I'm not spending an arm and a leg on food, rent and utilities I don't see this as being a problem. Being a “Mustachian" definitely helps).
* Not opposed to living overseas
Sounds like your are talking of my hometown, Perth, in beautiful sunny Western Australia. As the centre of Australia's mining and oil and gas industry it's the place to be for a young mechanical engineer! With a lower unemployment rate than Melbourne or Sydney and greater access to jobs that require your qualifications you are more likely to get a visa. (Re the visa most people come out on an employer sponsored visa called a 457. Permanent Residency (PR) visas are harder to get, so most people do a 457 first (the other advantage with these are the employer pays all your relocation costs, health insurance etc) and then apply for PR after 2 years. )
The surf is also great (and the weather warm enough you barely need a wetsuit!) and the beautiful big wave/ pr surfing comp mecca of Margaret River is just 3 hours away, perfect for a mustachian camping weekend get away. And with 4 weeks a year of annual leave as standard, plus 10 public holidays you'll have plenty of time to explore your new home.
For Kayaking the whole city is built around the Swan river so it is easy to live right on the river. If you want rapids the best ones are in a town called Northam 100kms from Perth. (The annual competition is called the Avon descent.)
Regarding bike friendly, more or less any neighborhood within 12kms of the city would make it easy to bike, so it really depends if you want to live on the river or the coast. Riverside neighborhoods like Maylands are popular with young professionals as they have good amenities (small bars, restaurants etc) and are close to the city (5kms). Great dedicated cycle paths as well so you don't have to ride in the road. You can buy a basic apartment in this neighborhood for $220k (sounds expensive, but wages for mechanical engineers start at 80k and move north quickly as you obtain experience. ) Or you could rent a room with 2 others in a house for about $160/week. (Expect to take home >$1000 a week even after tax and 401k are taken out of your pay). Scarborough is a more expensive and further from the city (12kms), but you could surf every day before work. Also popular with young professionals this beachside neighborhood has bars, restaurants, and all other amenities. There are plenty of other suitable (bike friendly/fun) neighborhoods, if you ever make the move message me and I can tell your more. (Must avoid outer suburbs - AKA anti-mustachian suburban wastelands!)
In terms of diversity, Perth is hugely culturally diverse, and as a result has awesome food, good ethnic grocery stores etc. The weather and the smallness of the city are great. The smaller city makes it friendlier, as you tend to keep bumping into the same people at the same places so it is easier to make friends.
COL, I kinda talked about it before, but to look at COL alone is deceptive. Yes Australian cities have a high COL, but Australian wages are some of the highest in the world, so it does balance. (As stated above it would be easy to get rent well under 20% of you wage, which is fine.... You can go lower than this if you are willing to live with more people and know people who have been in their houses a bit longer (rent is hugely sticky down here, it often won't go up until you move house). I have a friend who just rented out a room in her place for $100/week... in a neighborhood only 40mins WALKING from the city centre.)
Good luck!