First off, a confession: this is mostly a list of equipment I have, but I'm sure that it'll serve as a guide to what you should get if you're serious about electronics as a hobby.
A quick look at my workbench revealed the following:
Soldron 25W soldering iron
Various screwdrivers (flat-headed and star, of 3 sizes each)
Fluke Model 11 digital multimeter
Motwane analog multimeter
Aplab L3202 0-35V, 2A bench power supply
Tektronix TDS-220 oscilloscope
Of course, there were also great heaps of tangled wire (in a variety of colours), boxes of components, etc. I'll come to that later.
Without doubt, the workhorse of my equipment. It doesnt have to be a fancy iron, just one that does the job reliably. I use a spade-tipped iron, but you'll need a pointed tip iron for doing surface-mount work. A good Soldron-brand iron will set you back around Rs. 200, but the improvement in quality over other irons is worth it. You may also want to consider getting a soldering iron stand, I just hang my iron by the cord on the table's drawer knob. Don't ever leave the iron flat on the table, it's amazing how much damage they can do!
Update: I now use a pointed-tip bit for my iron (you can buy replacement tips at Rs. 30 each for Soldron irons), with rather mixed results. Firstly, they aren't as hard to use as I thought they'd be. They do not rapidly lose heat or anything, but the technique used while soldering takes a bit to get used to. You'll have to lean the iron sideways to heat the maximum possible area of the board. While tilting the iron this way, don't allow the solder to fall onto other pads. Aside from that, pointed-tip irons seem like quite a good idea.
Warning: Soldering irons are dangerous and can cause very painful burns. Handle with care, and dont hold me responsible for any damage due to use.
Along with the iron, get a big reel of soldering lead (with rosin core, NOT acid core) and a small tin of soldering flux. It may help to get two grades of soldering lead, with different thicknesses. If you've bought a new soldering iron, read the next section carefully.
Replacing soldering iron tips
After a while, your soldering iron's tip may get oxidized. Not only does this make the iron look bad, it also causes problems while soldering. If the oxidation is too severe, you can replace the tip quite easily. The old tip should come off after a hard pull or two. The heater element (the tubular section onto which the tip slides) must be coated before inserting the new tip. I'm told you need some special fluid for this, but I use ordinary heatsink grease. If anyone knows better, please contact me. Slide the new bit in place, but don't heat it yet, you must first coat the tip with soldering flux. After the iron heats up and the flux starts to smoke, evenly apply solder to the tip of the iron. This forms an even coating of solder over the entire tip. This step is absolutely essential, because if the iron is initially heated without the flux, solder will never "take" to the tip. Naturally, this should also be done with new soldering irons.
Mine are more or less generic, no big-name brands here. You could even make do with those "Multi-screwdriver kits" you can buy, since you usually wont be using them for heavy-duty tasks. Then there are the unusual uses for screwdrivers, such as spreading flux and heatsink grease, prying out DIP ICs from their sockets, etc.
These are useful for, well, cutting wire. Some people try to use them to strip insulation from PVC wire, with mixed results. You could try a Wire Stripper for this, I use my teeth :-). Also useful to clip component leads after soldering. They tend to go blunt after a while, so it makes sense to get two.
Nose pliers are good for all sorts of general uses. In a pinch, they can be used in place of tweezers, to hold small components while soldering (saving your hands from being burnt), to bend component leads, hold wire while tinning it, etc. A novel use is to hold transistor and IC cases while soldering (it acts as a temporary heatsink).
This is a must-have piece of test equipment. I have two, and yes, you may find you'll need two. The digital meter is pretty high-end, but it's my dad's, so the cost problem wasn't there. Cheaper DMMs from MECO, etc. are available for under Rs. 200. My DMM measures AC/DC voltage (not True RMS, tho), continuity, resistance and capacitance. I use the analog meter mainly because the DMM cannot measure current directly. Also, the DMM's maximum voltage is limited to under 1kV, while the analog meter can measure up to 2.5kV (yes, I used this range once!!!) Handle analog meters with care, since their movement is usually quite delicate.
Bench Power Supply
While a bench supply helps a lot, it's not essential. You can build yourself a decent supply for most purposes, anything more than 15V is usually overkill for most people. An LM317/LM337 based supply would be ideal for most people. I will post a good circuit for a power supply when I find one; it makes an excellent first project.
Oscilloscopes are the ultimate test instruments, but their cost (Rs. 10,000 for the cheapest possible 'scopes) puts them out of reach of most hobbyists. In case you know someone with a 'scope, occasional tests can be done elsewhere. College students can try things in their labs. Digital scopes are generally much easier to use, but have a price tag of Rs. 75,000+. Yes, I'm very lucky to have that 'scope <gloat!>