Nuclear fuel pellets - when does the chain reaction start?
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.
Adam Redwine does a splendid job of answering the question but I'd like to add just a few clarifying points. As stated, the neutrons created by fission are too "Fast" to cause a significant amount of fission. They need to be slowed by a moderator (water is what most commercial reactors use). But what prevents them from uncontrolled fission as soon as we place them in water? Two things, distance and neutron poisons. There is no "pellet activation" as seen in another answer. When the new fuel is ready to be put into the reactor (assuming we're refueling an existing reactor) it is placed in the spent fuel pool. The fuel bundles are spaced apart from one another to prevent inadvertent criticality (the state of self-sustaining fission) and the water contains boron. Boron absorbs enough neutrons to prevent the chain reaction from starting. The fuel is then kept underwater throughout the entire loading sequence to the reactor core. Once in the core and after the reactor is re-assembled, the control rods also absorb neutrons to prevent the chain reaction. When the operators are ready to start the reactor they carefully reduce the boron concentration in the water and withdraw the control rods to allow the neutron population to increase until the chain reaction becomes self-sustaining, which is a critical reactor.