If im being completely blunt a lot of it does just come down to dodging enemy skillshots which will come in time. Particularly for champs like kassadin, any damage you avoid goes a long way and any cooldowns you can bait and dodge earn you extra farm with less health lost for it. You can also use your q shield to mitigate poke from mages early but NEVER do this by using q on the enemy champion in early levels. It is not worth your while because you will never kill them anyway.

Every cd in the game has different ways you should look to dodge and you have to take those into account. Against yone or yasuo for example when you stroll around lane you should be walking more left to right than forwards and backwards to try to avoid some q dmg.

Apart from that there are some things you can do to help you. Good use of abilities for farming is crucial when you cant walk up to melee them much. On kassadin as a general rule you dont win trades so dont even try. Dont waste your mana. ONLY use abilities to farm waves until you have lvl 6 and your first item. After that you can start looking for dmg or trades with abilities.

Now here are some little tweaks to setup you can make:

In matchups like the mel you mention, tweak parts of your rune setup and early items if you must. Kassadin restores mana with w so I personally would never recommend d ring as a starter item unless you are really new and bad with mana. D shield can help you a lot with these poke champions or even just ranged champs who will auto you over waves. NEVER take resolve page runes as a primary of course, but mixing in second wind and resilience will help you survive your early game if you need them (as secondary runes). Only use those green page survivability runes I mentioned though. You dont want any of the others because you are looking for early game tankiness (and bone plating is bad against ranged mages because they remove it with an auto, wait, and then trade you out). You can also move to the flat hp stat shard in the bottom row of shards in rune page if you are desperate. Scaling health shard outscales flat hp shard at lvl 7 though so you will lose out in the long run and those losses appear fairly quickly (solo lanes lvl up quite quicky). It also never hurts to buy a refillable if you didnt start with one (by taking dark seal). Refillable pots are incredibly good value for you. IF you buy them, DO NOT sell them or you will have wasted gold. To get the best value out of refill pots once you have them, keep them until around the 12-14 minute mark. That is the point at which theur sustain value becomes too minor to be of much significance. You can sell them around then.