Storm Guide is a feature of the Storm Sorcerer from XGtE (p. 52):

At 6th level, you gain the ability to subtly control the weather around you.

If it is raining, you can use an action to cause the rain to stop falling in a 20-foot-radius sphere centered on you. You can end this effect as a bonus action.

If it is windy, you can use a bonus action each round to choose the direction that the wind blows in a 100-foot-radius sphere centered on you. The wind blows in that direction until the end of your next turn. This feature doesn't alter the speed of the wind.

Can Storm Guide stop rain and redirect wind created by magic?
And if so, does Storm Guide also stop the magical effects tied to the rain and change the direction of magical effects tied to the wind?

There are a handful of spells that create wind with magical effects, for example gust of wind, control winds, and whirlwind.
The only example I could find of rain that comes with magical effects is sleet storm.

In short, this feature will work on magically created weather

Features in 5e do what they say they do. If the feature was meant to exclude a particular category of weather (like weather created by a magical source) then the spell would describe those exceptions. Since the feature doesn't make any exceptions, if a magical effect creates weather, then this ability can affect it.

With that being said, there is one caveat...

It depends on the DM's definition of "weather"

In 5e, terms that do not have explicit game definitions are meant to be interpreted based on their natural language definitions.

Although a section on weather does exist in the DMG, it doesn't define weather as anything explicit in game terms. So we use the natural definition, which merriam-webster defines as:

the state of the atmosphere with respect to heat or cold, wetness or dryness, calm or storm, clearness or cloudiness

At first glance, everything looks fine. But colloquially, not every single atmospheric effect is what I would call "weather." And that's where your sticking point is.

Do these spells actually count as weather?

It seems a silly question, but think about it. If you use a leaf blower to clear your lawn, does that count as weather? If I stir the water in my glass to make a miniature whirlpool, is that weather? If not, what makes those that different from using magic to create these exact same effects only on a larger scale?

In fact, there is even some precedent that not every atmospheric effect is weather. When asked about the use of create or destroy water on a vampire's mist form, Jeremy Crawford, lead rules designer, replied:

Create or destroy water—the spell can destroy a 30-foot cube of fog. A vampire's mist form is not fog or any other weather effect.

A DM is well within their right to rule that one or more spells that have an effect on the atmospheric conditions around a character do not constitute weather. Taking a previous example, it would be akin to them saying "gust of wind doesn't create weather, it's just a magic fan blast. That's not weather." If they make such a ruling, you're out of luck. The feature only works on weather, and your DM said that particular effect isn't weather. However, the DM is just as justified saying "oh, yeah. That's totally weather." and then you're good to go.

TL;DR

Since there's no solid answer on this specific scenario from any official rule clarification (as far as I could find) the answer then boils down to: You might be able to as long as your DM rules that the magical effect really does constitute "weather" instead of magical manipulation of the atmosphere (like a fan on a hot day, only magical instead of mechanical).

Maybe as written, but it's probably not intended to

As Adam's answer points out, the feature as written might be able to affect magically created wind or rain - though it might depend on the originating spell/effect and weather it's actually described as wind or rain.

Perhaps more importantly, though, the Storm Guide feature is not intended to have a major mechanical benefit. On the last page of Unearthed Arcana: Waterborne Adventures, where a version of the subclass originally appeared, there's a sidebar below the Storm Guide feature describing "ribbons" and using the feature as an example:

Ribbons

On the R&D team, any ability meant to convey flavor rather than a mechanical advantage is referred to as a ribbon—a thing that’s mostly for show. Thieves’ Cant is a great example of a ribbon ability, and Storm Guide also falls into this category.

We don’t weigh ribbons when balancing one class or option against another. For example, Heart of the Storm carries the power load at 6th level for the storm sorcerer, while Storm Guide is here only to show how these sorcerers can excel as sailors. It isn’t meant to help in combat, but it’s potentially very useful in maneuvering a ship.

In short, Storm Guide is not intended to provide any significant mechanical advantage - it just helps guide the flavor of the class. You should keep this in mind when making rulings on how the feature works when it comes to magically created wind or rain.