A lot can be, and has been, written on the subject, but I'll give you the short and sweet version.
Does nuclear chain reaction start in fuel pellets even before they being installed in reactor? -- No
There are several reasons why this is so.
The number of spontaneous fissions of $^{235}$U is minimal. The branching ratio for that mode of decay is $7 \cdot 10^{-9} \%$, which means that for every billion $^{235}$U atoms that decay, only $7$ of them do so by spontaneous fission. This does not produce enough neutrons to start a chain reaction.
The neutrons released from fission have too much energy to induce many more reactions. The probability of an atomic event is characterized by the associated cross-section. The cross-section for the relevent fissions of $^{235}$U at the fission spectrum average is 1.235 barns. This is not zero, but it isn't very large; compare this to neutrons in the 0.025 eV range where the cross-section is 584 barns.
Fresh fuel rods are not typically enriched very much. The exact enrichment varies depending on a variety of factors, but fresh fuel is typically on the order of 2-5% $^{235}$U; most of the rest of the fuel is $^{238}$U which is significantly less likely to fission due to neutrons in the average fission spectrum.
As to your confusion, yes, the fuel is sufficiently enriched to sustain a chain reaction; that is what it is designed for. It is designed, however, to be inside a reactor when that happens. Inside a reactor, there are other things that start and sustain the chain reaction. The primary of these is a moderator.
A moderator is a substance that slows the neutrons down from the fission energy of around 2 MeV to the average temperature of the moderator, around an eV or so. In all commercial reactors in the United States, this moderator is plain old water. Some reactors, though, use heavy water and others use graphite. Either way, the function is critical for (most) nuclear reactors. There are such things as fast reactors, but I'll let you research that on your own.
Also, for fuel pellets and rods are radioactive, how do we transport them? -- Very carefully.
Fresh fuel rods, as explained above, are not dangerously radioactive and can be handled without a great deal of extra caution. Fresh fuel is made into pellets that go into rods; the rods are assembled into assemblies which are transported to the power plant in transportation casks. These are often times not much more than a wooden box with packaging material. They can be loaded on the back of a semi or onto a train and shipped to the power plant.
Spent fuel rods are typically moved by machine and only then a very short distance into cooling pools. These pools provide sheilding while allowing the removal of excess heat generated by fission products. After many years in a cooling pond, many nuclear power companies have started to move spent fuel to dry cask storage containers. These containers provide the same benefits of the cooling ponds, but require less maintenance.
Graphite in reactors gets radioactive mainly by forming beta decaying
$\require{mhchem}\ce{ ^{14}_6C}$,
mostly from naturally occuring stable $\ce{^{13}_6C}$ (1.1% abundance) :
$$\ce{^{13}_6C + ^{1}_{0}n -> ^{14}_6C}$$
with half-life $5730 \pm 40$ years. It is naturally present in traces in all carbon containing matter with recent carbon interchange with atmosphere.
The beta decay has the equation:
$$\ce{^{14}_6C -> ^{14}_7N + ^{0}_{-1}e + \overset{-}\nu_{\mathrm{e}}}$$
It is the same isotope used for radiocarbon dating, as it is continuously created in atmosphere by cosmic radiation.
Heavier carbon isotopes are very unstable and quicky undergo beta decay to nitrogen ( or even oxygen ).
Secondary source of radioactivity is contamination by fission products and radioactive products of neutron irradiation of stable isotopes.
For the former, there is wide range of formed radioactive isotopes, with 2 distribution peaks with relative masses about 2/5 ( like $\ce{^{90}Sr}$ ) and 3/5 ( like $\ce{^{131}I ^{137}Cs}$ ) of the mass of $\ce{^{235}U}$.
If you used graphite control rods, you would cause the 2nd Czernobyl.
RBMK uses graphite tipped control rods, not graphite control rods.
These tips cause problem in short reversal of control rod effects.
Graphite is not used as control rods, but as a neutron moderator, similarly as light or heavy water.
Moderators serve for slowing down neutrons to thermal speed(thermal neutrons) to improve the cross-section of the fission reaction, so the sustained fission reaction is possible even in lightly enriched or even natural uranium(heavy water nuclear reactors). Best moderators have low neutron absorption.
Moderation principle is on purely mechanical bases, as fast neutrons collides with slow particles (protons, deuterons, carbon kernels) and collisions redistribute momentum.
From mechanical point of view, protons would be best moderators, but unfortunately they considerably fuse with neutrons, so enriched uranium is needed ( in the opposite to deuterons )
Control rods are the opposite. They serve for regulation of "neutron economy" by absorption of excessive neutrons.
There are usually used 3 sets of rods,
Regulation set for BAU regulation
Adjusting sets, progressively pulled out, as neutron economy gets worse in the time
Emergency set, that stops the fission when triggered to be pushed in.
The rods contain usually cadmium or preferably boron, with high cross-section for neutron absorption.
Boric acid is also used in cooling water in secondary/tertiary circuits, where water serves as heat transfer medium only, not as moderator.
Best Answer
You are missing the concept of critical mass. In a small amount of uranium, like a single pellet, some fraction of the atoms will decay spontaneously every second. When enough of this is put in proximity, then the emissions of some of the decaying atoms kick other atoms just right so that they fission too. As a result, more atoms fission than you would predict from just the probability of a single atom doing so.
The main difference between a small pile of uranium and a large one is that in a large one you get a chain reaction. Power plant reactors rely on this chain reaction mechanism to get the large amounts of output power. The control rods control how much the emissions of fissioning atoms can hit other atoms, thereby controlling the overall reaction rate.
In reality, this description is over simplified. There can also be moderators envolved that sortof convert some of the fission results into stuff that can kick other atoms to fission when without the moderator they would not. However, that is a aside to this question.