No need to use ice unless you have swelling. Hot baths work for some but not all; give it a go and see if it works for you.
For muscle soreness the Physios and Rehab instructors at my place (Army) swear by foam rollers; hurt like hell but are really good at relieving muscle soreness. Lots of info on line about how to use them.
Painful shins need to be treated differently to muscle soreness as can lead to shinsplints etc. Best advice I ever got (sounds obvious) is to reduce the exercise that causes the issue until it calms down and then build up again slowly. You will not get better by gritting your teeth and cracking on, no matter now frustrating it may be.
As @dodgyknees said, shin splints are a nightmare as they’re usually caused by either running too fast or too much too soon or even both so you need to rest them then ease into the running! Have you made sure your running shoes are the right ones for you?
Also just make sure your laces aren’t too tight or loose, you need to find a happy medium and my problem was that i kept heel striking so just watch for that as well
With the muscle soreness have you made sure you cool down properly? A Bi product of vigarous training is lactic acid from where your muscles use glucose to generate energy, it’s kinda like the body’s defence system against using them too hard where you do literally feel the ‘burn’.
An effective cool down will help, something thats less intense than the workout you just did, so finish a run off with a 3 min slow jog into a 2 min walk, should help flush it away!
Also cold baths and ice do actually help against lactic acid! Good luck with your training