The Find Familiar spell is currently the only method for a wizard to obtain a familiar, and it lists the creatures available. However, a large number of creatures in the Monster Manual, such as the pseudodragon, imp, or quasit, include the 'Familiar' variant. Note that a Warlock with the Pact of the Chain feature can obtain many of these creatures as a familiar.
What you have to remember is that variant monsters, like monsters, are designed for the GM to use to make enemies more interesting. The Familiar variant is a monster variant, just like the troll's Loathsome Limbs variant or the Genie Powers variant. They're for GM use rather than player use.
The Mage NPC in Appendix B of the Monster Manual also has a Familiar variant, which says:
Any spellcaster that can cast the find familiar spell (such as an archmage or mage) is likely to have a familiar. The familiar can be one of the creatures described in the spell, (see the Player's Handbook) or some other Tiny monster, such as a crawling claw, imp, pseudodragon, or quasit.
So the Familiar variant is for GMs to create more interesting NPCs, rather than to provide players with additional options. Of course, with your GM's permission, you could obtain a more interesting familiar. This would probably involve actually finding such a creature and somehow forming a bond with it. But this relies solely on your GM to allow and arbitrate.
It's also worth considering that allowing a wizard to obtain one of these more powerful creatures somewhat invalidates the Warlock's Pact of the Chain.
No, a Warlock with the Pact of the Chain feature does not receive Magic Resistance if they choose a Quasit, Imp, or Pseudodragon. In the first place, a variant rule is only in play if the DM chooses. However, even if the DM decides that pseudodragon familiars (for example) are a thing, it still doesn't benefit the Warlock.
Some pseudodragons are willing to serve spellcasters as a familiar. Such pseudodragons have the following trait.
So "some pseudodragons" will be familiars who share Magic Resistance with their masters. The Warlock, however, gets their familiar from the find familiar spell, which says:
You gain the service of a familiar, a spirit that takes an animal form that you choose: [...]
The Pact of the Chain says that:
When you cast the spell, you can choose one of the normal forms for your familiar or one of the following special forms: imp, pseudodragon, quasit, or sprite.
In other words, the Warlock's familiar isn't any sort of pseudodragon, let alone one with the variant. It's a spirit that takes the form of a pseudodragon (and can take a different form anytime the Warlock chooses). The spell does say that:
[...] the familiar gains the statistics of the chosen form [...]
However, the statistics of a creature are defined on pages 6-11 of the Monster Manual. The short definition is that, if you look at a creature's entry, it's everything in the yellow box. This doesn't include variant traits like the Familiar trait - those are in green boxes off to the side.
All of that aside, a generous DM could, of course, allow a Pact of the Chain familiar to provide its master with Magic Resistance. It must be pointed out, though, that this is making the Pact of the Chain vastly more powerful than it normally is, so DMs thinking about it should consider carefully.
An even more generous DM could allow any player to gain the service of a creature with the familiar variant. This, too, should be considered carefully - it's effectively giving the player a more powerful version of the Pact of the Chain feature for free.
Best Answer
All of these creatures are in the Monster Manual. However, you can also find the ones that don't appear in the PHB in the Dungeon Master's D&D Basic Rules posted on the WotC web site.
You can also find the find familiar spell on D&D Beyond, freely available (as it is part of the Basic Rules), and from there follow links to the Basic Rules version of each of the possible familiars — like octopus.