Yes it does, although it become unnoticeable after a point. Newton's law of universal gravitation states Force = G * (m1*m2)/d2. Where G is the gravitation constant, m1 and m2 are the mass for the respective objects and d is the distance, squared. The effects grow weaker the farther the object is, to the point it doesn't matter, but as long as Force isn't zero --which it truly can't be -- a force is acted on the object(s).
It keeps them in orbit
The other planets do have gravity.
Because the planets still have gravity. Thing of an earthquake like a house settling. As gravity pushes down on the house, the house ever so slightly moves itself. The effect is amplified with planets; they are "settling" because of their own gravity.
The motion of the planets are governed by a number of factors: Gravity - The planets are being pulled towards the Sun in free fall. The velocity of the planets - This balances against the force of gravity opposing it and producing on balance the orbit. The mass of the planets. The effect of gravitational attraction between the planet and any satellites it has. The effect of gravitational attraction between the planets and other planets/dwarf planets etc.
Planets with elliptical orbits are affected by the difference in the strength of the Sun's gravity as they move closer to, or farther away from, the Sun. The only other major effect on planets are the gravity of other planets, notably Jupiter and Neptune. Except for Mars, Ceres, Pluto, and trans-Neptunian planets, the effect is extremely small -- it has resulted in stable orbital resonances between several small outer planets and the planet Neptune.
All massive objects (including all planets) have gravity.
The existence of large objects like planets or stars curve space and time. As the Earth goes around the sun it is in fact falling into the sun. The problem is that the sun and other planets are also moving. Gravity from the sun's previous location takes time to reach the Earth and planets such that the gravity from another location reaches the targets first. As the Earth and other planets reach a location where it might be moving in; the sun moves again, and other planets impart their gravity as well. The gravity of the sun becomes diluted throughout the heliosheath because of this weakening the effect of gravity even further.
my inability to go to other planets. and gravity
All planets are affected by the gravity of other planets.
Due to the moon's strong gravitational pull, relinquished by other planets, tides are formed.
because of the gravity of the other planets around them
Simple. Gravity! The planets are dragged in by the gravity of the sun.