How does a bicycle tyre and wheel affect its performace and speed? I have 700 x 38c wheels on my bike. While reading online I found that wider tyres have lower rolling resistance at the same inflation pressure than narrower tyres but they will be a little heavier and will offer more aerodynamic drag. I am not bothered about the weight because it won't make much difference if I am riding mostly on flat ground --  aerodynamic drag and rolling resistance drag are far greater sources of drag than that caused by the direct increase in weight of the tyre. 

Considering that, I am not sure which will perform better on the road. My online research shows that wider is better.

What are the other factors of a wheel (like untrue wheel) that contribute to performance of a bike.
The general answer is "it depends". Certainly, weight reductions help increasing speed, but improving the rider's strength and good bike fitting will go much further for improving performance than tire faffery. (Obviously, for professional racers, this advice doesn't apply - cutting grams and everything is part of what they need to do to win).

Wider tires should be run at lower pressures than narrower tires. It is true that rolling resistance drops with pressure increases and for similar tires of different widths at the same pressure the wider will have lower rolling resistance (but this is moot by the prior sentence). Note the word "similar" - a knobby tire of the same width will have higher rolling resistance than a slick. 

Wider tires can better absorb bumps and what not, so they can increase control giving net better performance in the real world. However, as you see on many road race bikes designed for purely nice pavement, thin tires are normally spec'd. 

As for aerodynamic drag, the paper "Aerodynamic Characteristics of Low-Drag Bicycle Wheels", Aeronautical J., Vol. 99, No. 983, Mar. 1995, pp.109-120 (summarized here) says ~10-15% of the drag is due to wheels, and by switching wheels you can get net 2-3% improvement. This is tiny, especially when you look at how much fancy aerodynamic wheels cost (and their low durability). For someone who is not racing, its almost surely better to put the money you might spend on a fancy aero wheel into having a durable wheel. 

Now, to the reality of the situation: A wider tire is going to give you better ride quality and you can get very good rolling resistance by picking a smooth tire for road use. You have to balance durability of the tire and the wheel if you choose to do improvements here. For most people who are not in race situations, pick a decently wide tire, play with the tire pressures so you get a good ride and low rolling resistance and move on with your life. In theory, depending on the terrain, you might be able to go faster with a narrower higher pressure tire, but the possible ride and control effects may hurt you in net.