Willk's answer says that the Moon is about 1,000,000,000 times as massive as Deimos or Phobos. A moon a thousand times as massive as the Martian moons would be one millionth as massive as the Moon. A moon a million times as massive as the Martian moons would be one thousandth as massive as the Moon. If moons of those masses were at the same distance as the Moon is from Earth, their tidal effects would be miniscule compared to those of the Moon on Earth.
Of course the strength of their tidal effects on the planet will depend on how close or how far they orbit as well as on how massive they are. Presumably different moons would orbit the planet at different distances.
Any permanent moons of the planet that keep on orbiting it for many millions and billions of years would have to orbit within it's Hill Radius.
The Hill sphere of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere.
In the Earth-Sun example, the Earth (5.97×1024 kg) orbits the Sun (1.99×1030 kg) at a distance of 149.6 million km, or one astronomical unit (AU). The Hill sphere for Earth thus extends out to about 1.5 million km (0.01 AU). The Moon's orbit, at a distance of 0.384 million km from Earth, is comfortably within the gravitational sphere of influence of Earth and it is therefore not at risk of being pulled into an independent orbit around the Sun. All stable satellites of the Earth (those within the Earth's Hill sphere) must have an orbital period shorter than seven months
The Hill sphere is only an approximation, and other forces (such as radiation pressure or the Yarkovsky effect) can eventually perturb an object out of the sphere. This third object should also be of small enough mass that it introduces no additional complications through its own gravity. Detailed numerical calculations show that orbits at or just within the Hill sphere are not stable in the long term; it appears that stable satellite orbits exist only inside 1/2 to 1/3 of the Hill radius. The region of stability for retrograde orbits at a large distance from the primary is larger than the region for prograde orbits at a large distance from the primary. This was thought to explain the preponderance of retrograde moons around Jupiter; however, Saturn has a more even mix of retrograde/prograde moons so the reasons are more complicated.2
So any stable long term satellites of Earth would have to orbit within about 500,000 or 750,000 kilometers of Earth, within the true region of stability. Unstable short term temporary satellites of Earth could orbit within the outer parts of the Hill sphere of Earth, at distances of about 500,000 to 1,500,000 kilmeters, and even at distances beyond the Hill sphere. The farther from Earth their orbits are, the shorter their periods of being satellites of Earth is likely to be.
The size of a planet's Hill sphere depends on the mass of the planet, the mass of the star, and the distance between them. If your planet is exactly as massive as Earth, and your star is exactly as massive as the Sun, and the semi-major axis of the planet's orbit is exactly one Astronomical UNit (AU), the size of the planet's Hill sphere will be exactly the same size as Earth's Hill sphere.
If any of those factors differ significantly, the size of your planet's Hill sphere will also differ significantly from that of Earth. In that case you might need to calculate the size of your planet's Hill sphere.
If some or all of the moons need to be visible as objects instead of mere dots from the planet, the visual acuity of the Human eye will determine minimum diameters that they must exceed (unless your characters are all aliens with different eyesight than humans).
The maximum angular resolution of the human eye is 28 arc seconds or 0.47 arc minutes, this gives an angular resolution of 0.008 degrees, and at a distance of 1 km corresponds to 136 mm. This is equal to 0.94 arc minutes per line pair (one white and one black line), or 0.016 degrees. For a pixel pair (one white and one black pixel) this gives a pixel density of 128 pixels per degree (PPD).
A full circle is 360 degress. 28 arc seconds is 0.008 degress, so there are 1 divided by 0.008, or 125, times 28 arc seconds in a degree. Thus there are 125 times 360, or 45,000, times 28 arc seconds in a full circle. 28 arc seconds is 0.0000222 of a full circle.
The radius of a full circle is about 1 divided by 2 divided by 3.14159, or 0.159155, of the circumference of that full circle. the circumference of a full circle is about 6.28318 times the radius. Thus the width of an object 28 arc seconds in angular diameter should be 0.0001394 of the radius.
So an object with an angular diameter of 28 arc seconds should have a physical diameter of 69.7 kilometers at a distance of 500,000 kilometers, and a physical diameter of 348.5 kilometers at a distance of 2,500,000 kilometers.
And possibly the mini moons would need to be a few times 28 arc seconds in angular diameter to be seen as extended objects instead of as dots of light.
And if you want the small moons to appear as mere dots of light they can be much smaller.
Jupiter has four large moons, Io, Europa, Ganymede, and Callisto. And if they were farther from the brightness of Jupiter they could be seen by the naked eye, without a telescope. And possibly they have sometimes been seen with binoculars or a telescope.
All four Galilean moons are bright enough to be viewed from Earth without a telescope, if only they could appear farther away from Jupiter. (They are, however, easily distinguished with even low-powered binoculars.) They have apparent magnitudes between 4.6 and 5.6 when Jupiter is in opposition with the Sun, and are about one unit of magnitude dimmer when Jupiter is in conjunction. The main difficulty in observing the moons from Earth is their proximity to Jupiter, since they are obscured by its brightness. The maximum angular separations of the moons are between 2 and 10 arcminutes from Jupiter, which is close to the limit of human visual acuity. Ganymede and Callisto, at their maximum separation, are the likeliest targets for potential naked-eye observation.
The smallest of the Galilean moons is Europa, which has a diameter of 3,121.6 kilometers. The closest distance between Earth and Jupiter, is about 4.2028 Astromical Units. Since an Astronomical Unit is 149,597,870.7 kilometers, Europa can get as close to Eartj as 628,729,931 kilometers, give or take a few million.
So Euorpa could be visible to the naked eye as a dot of light, if not hidden by the glare of Jupiter, at a distance which is 201,412.7149 times the diameter of Europa. A full circle at the closest distance between Earth and Europea would be about 1,376,011.386 times the diameter of Europa. The diameter of Europa is 1 divided by 201,412.7149 times a distance at which Europa would be visible to the naked eye if not lost in the glare of Jupiter.
So an object 2.482 kilometers in diameter should be visible as a dot of light at a distance of at least 500,000 kilometers from Earth, and a object 12.412 kilometers in diameter should be visible at a distance of at least 2,500,000 kilometers from Earth.
And that is a rough calculation, not allowing for the different albedos of different astronomical objects and for the more intense sunlight at Earth's orbit, nor for how much Europea exceeds the absolute miniumum apparent magnitude.
But it does suggest that your small moons should be at least a few kilometers in diameter to be visible even as points of light, unless they are a lot closer to your planet than the outer regions of the Hlll sphere, where temporary moons would usually be expected to orbit.
And I think that a diameter of a few kilometers is a lot larger than the diameters of any known temporary moons of Earth. All known temporary moons of Earth have been only a few meters in diameter.
That would suggest that your planet's orbit would have a lot more fairly large size asteriods with similar orbits than Earth has. That suggests that major asteroid impacts and their effects should be a lot more frequent on your world than on Earth.