i hope i can provide some help in assasin matchups. Ill split this post into several parts
Part 1: itemizing against assasins in lane.
if you dont feel playing against assasins as a mage you should start with dorans ring and 2 pots. that will get you some health to survive an all in, some ap and some mana regen. At your first back you should pick up another ring and a resist item (cloth armor or null magic depending on the opponent). You want that flat resist armor because your opponent will probably build some flat pen, and a resist item greatly mitigates that(remember that at low resist counts both pen and getting more resist are more useful). then it varies according to the type of the opponent you are facing. Against Ad assasins you want to finish your armguard, but dont go for the zhonyas, since armguard already gives you survivability. If you feel you need more survivability go for RoA(both swain and ASol use this great), if you dont get your first mana/cdr item, then continue building normally until you feel you need to finish the zhonya. Against APs you can either start building RoA if you are scared, or finish merc treads and your mana/cdr item if you arent.
You need to know that with this build you will come online later than usual, but the point is to outscale the assasin anyways.
TLDR: 2 dorans and resist items are a good buildup vs assasins
Part 2: laning against assasins.
There are several things to remember when laning vs assasins so here i will give some tips in no particular order to remember.
First off, dont stand near your minions when you cs/harrass, because assasins have want to hit both you and minions with abilities so they can kill you without being showed in. Always stand a bit to the side of your minions, since for example then zed will have to use his shadow and shuriken to clear the wave OR hurt you, but he cant do both, similair with kata daggers and talon w. the second thing is to know the assasins kill pattern. for example you know zed ult will make him appear behind you, same with shunpo, fizz e will jump right on top of you, and talon will q right in your face. Use that knowledge to quickly disable them with a veigar baby cage, swain claw, ASol ulty or raka equinox.
The second thing to know is to realize that assasins actually have two engage ranges: the full damage one and a reduced damage one.
the reduced damage range is a range where an assasin can all in you, but he uses his damage tools to close the range. For example if a zed has to use his shadow to get in range to ult you, or if fizz has to use his e into q to start a fight, or if talon cant q from close range. At that range they certainly cant kill you from full health, but watch out if you are lower and use your range advantage accordingly.
The third and related thing is to count assasin cooldowns.
You should count their cooldowns to see when you can safely farm and harrass.
The fourth and hardest thing to learn is when to stand your ground.
There are cases when you can actually turn an all in around. If you are full health and they arent and they try to all in you, if you think you can kill them dont start running, because you will lose the trade if you just run and let them hit you. Several of the champs you play are also actually quite good at standing their ground in some cases, for example if you are soraka and they are all inning you on your equinox, they cant actually do much.
also to note Swain,ASol and veigar can take aftershock and greatly reduce burst from assasins, just think about it