I've just started playing for the first time. Does the ground fighting with the robots play a big part in the game or can the game be played without it?

Completely optional, as mentioned. If you want to have the option to play ground missions, then you MUST complete the ground mission training scenario at the beginning of your game. If you don't complete that training mission then you will never be offered ground missions throughout the rest of the game for that character.

As others have mentioned the ground battles are completely optimal, although it should be noted that these missions pay really well. I personally really liked the ground missions (for what amounts to just a minigame the mechanics are actually pretty well done), but if you just really don't care for that kind of RTS game then you can certainly skip it without any penalty (aside from not having them available for earning credits).

Ground battles aren't that hard. Here's my tips.

Only make one type of robot. Give him 1 repair beam and 3 rocket launchers. Give him the fastest feet, a mortar, and the head that blocks stunners.

To fight, select all of them and manually move them around the enemy robots. Keep moving them back and forth. You will get the hang of it eventually. As each robots rockets come to bear they will fire their rockets. If not firing, they will repair the robots around them.

If you have more resources available, you can try making robots with 1 repair beam, one stunner, and 2 plasma guns. Those are good too, but they are more expensive.

And when you capture a base make sure to build rocket launchers. They do the most damage.

If you have enough robots, make two groups. You make a group by selecting them and then pressing Control-1 (through control-9). Then press the number to select the group.

I hope this helps and you start to enjoy the ground battles as much as I do.

RTS missions definetly pay very well as others have said. I usually do them if I'm short on money. Or if the mission is fun, not too long and fairly easy.
One small thing some may not know is that you can also get RTS missions from planets with normal reputation. Not a huge thing, but nice to know.

Learning to manually control 4xlauncher/plasma, antigrav, firerate/speed head gives you huge advantage as you can engage enemy turrets and robots from maximum range of your weapons. You can easily destroy even large groups of enemy bots by getting in range, then backing away while keeping the enemy bots just in range of your launchers (great for crowd control).
When possible, go for missile turrets as defences. Still, bots should be your main priority. Turrets are there to sponge up incoming fire while your bots do the damage.

Some of the later RTS missions are very rough, almost puzzle-like. One that I can remember is the one where you are inside an equentor with very limited bots. Very tricky.

I'd say practise them and get good, the pay is amazing, and I enjoy breaking up the flying round :)
Manual control is god. Seriously, you can put all your bots on one flank and have one firerate bot on the other and manual control it, and hold against nearly non-stop waves with a bit of effort :P

I though it was the best part of the game. Definitely better than that pathetic mess of space combat...

To space combat, play it turn by turn, definitely. Don't let the autopilot do combat for you unless you really outgun someone and also outfly them (like, more than about 1.5 times their flyspeed), otherwise it will make stupid decisions that you could avoid.
For one-on-one combat, hover the mouse over your target after EVERY round, to see if they're still engaging you, or heading off. If they're breaking and running, you can choose to chase them if you're looking for a kill or a surrender, or break it off if it was self defense and you're not sure if you really want to keep it going.
Never one-on-one someone who can outfly you by a decent margin unless you can take them in a turn or two (any more than about 50 a turn is dicey, and above 100 is basically pointless).
Never one-on-one if you can avoid it>USE YOUR COMMS. If its a particularly annoying pirate, you may find that you can call every other ship in the system down on it, leading to it being hit by so much more than you alone.
For melee's such as those that normally result with Dominators round, you can either go for solo kills if you have the weaponry, or go for fast kills that you might get less credit for if its really tight, by checking on your allied ships to see who's shooting who.
Generals: Missiles/torpedo's are for killing almost purely>the fact that you probably have 2 or 3 rounds worth in the air means that if you want to let someone live, you'll have to turn off firing them turns in advance (which can be so much less intimidating :P).
Long range stuff is useful if you have a slow ship, so you can actually hit stuff more than once, but if you're fast and keep track of movements, short range weapon classes tend to have a lot more bang for your buck (see fragment cannons, they can match damage with some of the mid-level weapons even).
SO off topic so I apologise ;)