A brief disclaimer first: I do not own the alpha of GCIII, having bought GCII Twilight of the Arnor mere days before the announcement of GCIII. However, I have seen videos of the GCIII alpha being played and believe that the following suggestions are still applicable. However, I believe that these suggestions would allow Stardock to make GCIII the best strategy game that has yet been devised. So far, GCIII looks excellent and has great potential.

Not all of these ideas are entirely mine. Some of them I had heard about before (like citizen races and cultures) and expanded upon, others (like the politics system) I thought I developed independently and noticed only after the fact that they had previously been mentioned.

1. As stated in the title of this post, multi-level maps should be implemented. It is simply not realistic to show planet-to-star distances on the same scale as star-to-star distances. Therefore, one should be able to traverse the galaxy on a "galactic map", showing just stars and starbases. In order to colonize planets or mine asteroids, one should have to zoom in on a star and arrive at a system map. To explore anomalies, construct starbases (which show up on the galactic map), or do battle in deep space, one would have to zoom in on a tile of the galactic map.
2. Expand the multi-map system by dividing the propulsion branch into two lines - one starting with Hyperdrive (and continuing with Warp Drive, HyperWarp, Fleet Warp Bubbles, Stellar Folding) and used for inter-system travel on the galactic map, one starting with Traditional Rocketry (continuing with Ion Drive, Plasma Drive, Fusion Impulse Drive, and Antimatter Drive) used for travel within star system or within the zoomed-in areas of deep space described earlier. Both will be important; the first will allow a player to send an invading force to enemy territory more rapidly, while the second will allow the player to evade the enemy to repair during battle.
3. Increase the role of starbases in logistics significantly. Even as higher levels of life support are researched and/or the "ship ranges are unlimited" event comes up, travel should be very slow more than a few galactic-map hexes away from a starbase, since the warp bubbles that can be created by a massive starbase tens of miles in diameter are much smaller than those created by even the largest spacecraft. Fleets with bigger craft should of course move faster than those with smaller craft outside a starbase's area of effect, but never as fast as possible within the area of effect of a starbase. Such limitations would not apply within star systems, of course.
4. Influence should be modified, since the current mechanism of holding sectors makes no sense when applied to deep space. Instead, influence starbases that are located along trade routes should allow one's culture to be spread to the other involved planet (and ultimately civilization) through the usual modules. Additionally, players should be given the opportunity to subsidize trade routes to specifically carry items that will spread the culture of the subsidizing civilization. Influence through trade should also spread based on the cultural influence of the planets at the ends of the trade routes.
5. Continuing on influence, each planet in the galaxy should have racial and cultural characteristics based on who has influence there, what the trade lanes are like, etc. For example, a Terran planet that has traded extensively with a more populous Drengin planet and a lower-population Iconian one may end up being 55% Drengin, 30% Terran and 15% Iconian racially, but if the Terran Empire maintains cultural control it may be 80% Terran, 14% Drengin and 6% Iconian culturally. Racial and cultural diversity should be beneficial in some ways (greater tourism income, greater cultural influence, better relations with original state, improved morale) but harmful in others (temporary xenophobic trends, race riots, sometimes refusal to enter military service if at war with home empire). Evil empires should be given the opportunity to enslave other races or suppress their culture.
6. The area effects for all types of starbases should still apply to those that are located within star systems, or more generally to all of their effects on the lower-level maps (since starbases are ultimately constructed on said maps, even if they are shown on larger scales).
7. The ability to choose, in addition to a governmental system, an economic system (Slavery, Capitalism, Socialism, Communism, Hive Society), each with different benefits and with different starting races. For example, the Terrans and Korx would be capitalist, Yor and Thalans would have a hive society with near-total government economic control, some Good civilizations (Altarians, Drath, Torians) would be Communist with government control over large enterprises, the Iconians and the Krynn would be Socialist with worker control of large enterprises, and the Drengin and Korath would have a slave economy by default. Each system should bring various benefits and maluses and (esp. with regards to Slavery) should influence ethical alignment.
8. With a Star Republic, Democracy, and Federation, planets should hold local elections. Under a Republic the planetary government would be able to appropriate 25% of tax revenues collected by the state, under a Democracy 50%, and under a Federation 50% but with the ability to set an additional tax rate of its own on planetary citizens. Money earned by planetary governments will be invested in policing, education, planetary infrastructure, and social welfare, increasing morale and raising tax revenues over the long run but reducing access to money that a player may want to spend a certain way.
9. Normal planet quality should go up significantly, and should start from the upper tens and going well into the mid-hundreds. It should also be possible to set up internal trade routes within one's civilizations that transfer food and industrial capacity between planets and spreads influence as well.