As a triathlon coach I disagree with the advice to run on your whole foot. Although running shoes imply (through excess padding) that we should land on our heels, in fact the calcaneous bone is not meant to take the constant pounding. On the other hand the mid-forefoot is made up of flexible joints which respond to varied terrain as well as shock absorption. This method of running has been studied and developed over the past 5-10 years and most elite runners run on a mid to forefoot.
Yes, there is a bit of strain placed on the calf, but with gradual technique shift and stretching, it will disappear.
As far as having more endurance, you need to build up gradually. As a weight bearing sport the risk for shin splints etc is high especially if you place too much strain on muscles not used to the pounding. It is said that slow is smooth and smooth is fast...that means that to begin with you should concentrate on slow runs where you watch your form and cadence. Ideal cadence is between 160-180 foot strikes per minute. It will feel like you are taking baby-steps at first but once you get used to the increased cadence speed will be easier to build.
Also, watch running on concrete all of the time. Trail running is great for balannce and ankle strength and allows legs some downtime from the hard surfaces. I do about 70% of my running on trails.
If your shins are sore, ice them and watch for continued stress. If you just have slight muscle soreness, you might run but slowly (see above). Tendonitis and shin splints start slow but if not treated will have you out of the count in no time. Also going out for a light spin on the bike or trainer is a great recovery for sore legs...or if you have access to a cold stream/lake in which to soak them for 10-15 minutes.
Feel free to ask any more advice.
Good Luck!