In early-mid game, how to fight as a mage with low magicka regen?

It's really, really hard to play as a ranged character in Skyrim, since it's pretty difficult to keep foes at range for any length of time. There are some Paralyze spells later in Alteration, but in general you should plan on being able to survive in a melee for at least a little while.

Here are some suggestions to get you started:

• In the Alteration tree, there are spells that grant you armor, like Oakflesh. This can augment the lighter armor you're going to wear.
• Learn Alchemy and make potions! Potions that restore your health and mana should be chugged at every opportunity. Making them using Alchemy is much, much cheaper than buying them.
• Do some light Blacksmithing - even if you don't want to "get buff" as a Blacksmith, the ability to improve light armor can be a real bonus. Every armor point counts at the lower levels.

Some easy potion recipes to get you started include:

• Restore Health - Blue Mountain Flower + Butterfly Wing, both are in ample supply around Whiterun or at any alchemy vendor.
• Restore Magicka - Elves Ear + Red Mountain Flower, these ingredients are a bit harder to come by, but still pretty common among vendors.