The problem arises from a misunderstanding of the various types of actions available to you during your turn in 5e. During your turn, you get:
Movement - During your turn, you can move up to your speed (30 feet, unless your race, wearing heavy armor, or some other feature modifies it). This can be broken up into several segments if you desire, such as moving before and after attacks.
One Standard Action - the "thing" you do during the turn. PHB p.192 details the normal actions you can take, such as Dash (make an extra move), Attack (perform some sort of attack; although it's a single action, this might involve multiple strikes, depending on your abilities), Cast a Spell, and so on.
Up to One Bonus Action - IF you have an ability that grants you a bonus action opportunity, like the Rogue's 2nd level "Cunning Action" feature, you can take a bonus action on your turn to perform one action that you have access to. Even if you have multiple abilities that grant bonus actions, you can only take a single Bonus action per turn.
Up to One Reaction - the Reaction is a special action that is initiated only when triggered by the actions of another character or situation. Most commonly this is the Opportunity Attack, taken when another character drops their guard to move away from you. Like the Bonus action, you only get a single reaction, and if you use it for some reason, you don't get another until the start of your next turn.
Other Action(s) - generally, simple actions that takes almost no effort. As part of your movement and action, you can make simple interactions with objects around you. See the table on PHB p.190 for some examples.
Having cleared these up, this is the procedure for making a "Touch" Attack with your Familiar.
(I will use Wizard here because that's the class I think of first, however it is interchangeable with any other class that obtains a familiar.)
First: The Prerequisites.
The Wizard must have an action ready, either by it being their turn, or having prepared an action to trigger when the familiar approaches the target.
The Familiar must have a Reaction ready, by not having used one since the start of their latest turn.
The Familiar must be within 100' of the Wizard, and within range to make a touch attack against the target (typically 5'/the adjacent square).
If all that works out...
On the wizard's turn, they cast the spell.
The Wizard's active turn is put "on pause" here.
(Still) On the wizard's turn, their familiar uses their "reaction" to make the attack. This is essentially an unusual type of "opportunity attack". Note that the familiar does not get a full turn. This means no move, no bonus actions, no standard actions, etc... Only the spell-based attack.
If necessary, make an attack roll as per normal with the spell (using the wizard's stats).
Resolve the spell as normal, as per the wizard casting it, ie; failed attacks miss, roll saving throws, damage, etc.
The familiar's reaction is now done. It has used its reaction, and will not get another until the start of its next turn.
Play returns to the wizard, who can now continue with any further actions they choose to take.
Regarding the form of the familiar, it is "a spirit that takes an animal form you choose: [list]."
In previous editions of D&D, there were many options to get a more powerful familiar. The words "form you choose" seem to leave plenty of room for other familiar forms, and as long as those options are made available to any caster who obtains a familiar, I can't see it as being a significant balance issue.
"Just remember, anything that you guys can do, my npc's can try as well." - a DM's motto for balancing player power.