Arguably the most important checks you can make are with your brakes. Start with the pads; do they show signs of excessive or uneven wear? Check the discs for any signs of damage or cracking. Are there any cracks or leaks coming from the brake lines?
Before heading out, test your front and rear brake independently. Listen for any scraping, squeaking, or rubbing and make sure your bike stops smoothly. Also have a visual check of your levers/pedals to ensure no signs of damage and they all have the correct free-play. If you think anything does not look or feel right, have a certified mechanic check over your machine.
Next, have a visual check of your forks and shocks. Ensure no oil leaks are present and that each is free from damage or pitting. With the brakes on, press down on the front forks and make sure they travel, but not excessively and there is no notchy or clunky feel to it. Having someone help you hold the bike, press down on the rear suspension, again is there smooth movement? If not, inform your mechanic.