how blocks organised periodic tableWatch
Period number = number of electron shells
Group 1 = alkali metals, more reactive going DOWN the group because the outer electrons are further from the nucleus and so there is less attraction and it is easier to lose the outer electron
Group 2 (sometimes numbered as groups 2-12) = transition metals. They have various properties.
Group 7 = Halogens, VERY reactive. More reactive going UP the group (in fact, fluorine, F, is so reactive it is not allowed in schools). This is because at the top of the group, the outer electron is closest to the nucleus so the force or attraction is greater and so electrons are more easily pulled towards a fluorine atom.
Group 8 = Noble gases, not reactive at all (we call them inert, which is a synonym for unreactive) This is because their outer shell is already full.
Be careful when studying the actinides. They may have d-block behaviors, but because they also behave like f-block elements they are classified as being in the f-block not the d-block.
For individual groups the elements within them show trends in their properties and behaviors. Transition metals and f-block elements are normally left out of groups because the trends between them aren't so linear and many exhibit anomalous behaviors. What JustJusty said is mostly correct, but be careful not to call the group 2 elements transition metals*. They're not transition metals, and have their own trends, much like the other groups. They even have their own name - the alkaline earth metals.
These are some of the most reactive metals due to them having only one electron in their outermost shell. They become more reactive as you descend the group because of an increased distance between the nucleus and outer shell and (for students below undergrad level) an increased amount of shielding as there are more inner electrons which offset the nuclear charge. Because of this they can easily lose their outer shell electron. Another way of phrasing this is that they become more electropositive as you descend the group. This can be demonstrated through their reactions with water. Lithium will float, effervesce violently and be propelled around the surface of the water by the evolved hydrogen. It's an exothermic reaction and is very fast. Sodium will do the same, but much more violently and releases more heat than lithium. Potassium is where the real fun starts. It will do the same as the above two, much more violently though. It also releases so much heat that the potassium melts and forms a spherical shape. The heat is also enough to ignite the hydrogen produced in the reaction, which causes a lilac flame to form above the potassium ball. The lilac colour comes from the presence of K+ ions, which when in an excited state release visible light photons of that wavelength. Rubidium does the same as potassium, but produces a red flame and doesn't float. Caesium gives a blue flame and also doesn't float. The reactions become increasingly explosive as you descend the group. In general the density increases as you descend the group (with the exception of sodium-potassium) and resistivity increases (with the exception of lithium).
These are also highly reactive metals. They have filled s subshells in their outermost shell and are therefore more stable (and less reactive) than their neighboring alkali metals. They show the same trend in reactivity as the alkali metals for the same reasons as the alkali metals.
Thankfully you don't need to know much about groups 3-5 properties. They can be an absolute nightmare to study - especially boron.
Reactivity decreases as you go down the group. Yes, dioxygen in its ground state is inert, but that's an anomaly in the trend. This is caused by an increasing distance between the outer shells and nucleus and a decreasing effective nuclear charge, so it becomes harder for the atoms to attract stray electrons or electrons from other atoms. The melting and boiling points increase as you go down the group, and the structures become more complex. Most of this group is extremely toxic too, even oxygen! Sulfur isn't too bad, aside from its smell, but the others can be really damaging. Most of the group are quite powerful neurotoxins. Oxygen causes seizures, brain damage, memory loss, comas and eventually death at high concentrations. Ironically, the way in which it poisons you is also how it keeps you alive. Selenium and tellurium cause mania in the victims of their poisonings. They're like a real-life Sheogorath.
Another group of killers. Their reactivity decreases down the group due to the same reasons as in group 6. These elements are mostly highly reactive and are photosensitive, in that they react when exposed to sunlight. Fluorine is banned in schools like JustJustly said due to it being so reactive. Chlorine and fluorine are the chemical weapons of the 2nd world war (minus pumpkin gas and some others). They dissolve in water to produce a mixture of acids. In the case of chlorine these would be hydrochloric (HCl) and hydrochlorous (HClO) acids, so you can imagine what happens when they're inhaled. It's nasty stuff.
These are the most gentle elements in existence. They're almost all completely inert (argon-radon can form octahedral halide compounds, though this is very hard for them to do) and the few reactions that do take place get harder to do as you go up the group. The reason they react occasionally as they get heavier is because of the increased distance between the outer shell and nucleus and the decreased effective nuclear charge. Plato would absolutely adore these elements, as they are essentially in his "perfect state." They're so inert because they have a full octet in their outer shells.
Periods show the number of electron shells in each atom. So every element in the 2nd period would have 2 electron shells.
*I just realised JustJustly was talking about the IUPAC nomenclature, but calling group 2 a transition metal group is still wrong. The transition groups start at group 3 when using IUPAC names.