The star's overall gravity does not increase, the star just gets denser. Since gravity is stronger the closer objects are, when the star shrinks gravity is now stronger between all parts of it since they are now closer together. If the Sun were to collapse, the gravity exerted on itself would be greater since it is much denser and smaller, but the orbits of the planet would not change because they did not get any closer to the Sun.
The core of the star resists collapsing thanks to the outward pressure generated by the temperature of the gas/plasma. During the star's main sequence phase, fusion serves to replace the lost energy radiated away from the core, keeping the temperature of the core stable. Otherwise it would would go through a continual process of shrinking, heating up, losing energy, shrinking, and heating up more.
This may seem a bit counter-intuitive, but the core is a volume of hot gas under pressure. It resists the inward pressure thanks to the outwards force it gets from being hot. If energy is radiated away from the core, then it drops in temperature and loses a bit of outward force and the inward pressure compresses it, causing it to shrink. Since compressing a gas causes it to heat up, the core ends up with a slightly smaller volume and a slightly higher temperature than before. The temperature is higher than before because the star shrinks and the inward pressure due to gravity increases (because the star is now denser), requiring the core to be at a higher temperature to exert enough force to counteract this higher pressure. This is a continual process, the core undergoes all phases of this process simultaneously. The net result is that, without a way to replace this lost energy, the core gradually shrinks and grows hotter.
During main sequence, the energy lost from the core is replaced by fusion, so the temperature never falls and the core doesn't shrink. (Technically it does, but this takes place over a few billion years instead of a few million and is due to gradual loss of hydrogen fuel in the inner parts of the core) Once the fuel runs out the core undergoes a slow collapse like I explained above until the temperature is high enough to fuse helium and other elements (if the star is massive enough).