You need an antenna that receives signals from all directions - called omnidirectional.
Materials make no difference: any type of wire is good, as long as it keeps its shape. Metal coat hangers are popular, multi-stranded or solid makes no difference. I'm a fan of No 10 house grounding wire.
You need to connect your antenna to your RTL stick: coax cable is needed, read my hardware guide for details.
Recommended beginner antenna
A monopole with two radials optimised for 120-130 MHz - three 60 cm (2 foot) elements in a peace sign / Mercedes emblem shape, coax conductor connected to vertical element, braid to two sloping elements.
Used this antenna for Airplane position signals (1090 MHz) and AIS with success.
Construction details are left to your inventiveness, or...
the following is a quick, ugly and working solution:
Drinks bottle - neck of bottle should be sloping downwards.
Cut a small hole in the plastic and push the two bottom wires on each side through, so the ends come out at the top.
Connect these two wires to the braid of the coax.
Wrap the center conductor around the third wire, then wrap it tightly
Tape the third (vertical) wire to the drinks bottle.
Granted, it looks hilarious on pictures, but it gets the job done.
Compared to a commercial discone, the 80 USD Scanmaster, airport info voice sounds the same; because the discone picks up more noise the signal-to-noise ratio (SNR) is the same.
A more aesthetically pleasing solution is using screw terminal blocks and wood - this also enables quick disconnecting of antenna elements. The example above is just a proof of concept, demonstrating that household items can be utilised.
Put your antenna outside
The illustration on the right shows three scenarios:
1) antenna on windowsill indoors,
2) antenna on windowsill outdoor,
3) antenna 1 metre from window outdoors.
Signal is the height of the peak, red horizontal line added for easier identification.
Lessons learned: 1) Signal 8 dB less indoors - that's a lot. 2) No point placing it 1 m from window, windowsill is fine.
Even if you live in a condo / flat with restrictive antenna policy, place the antenna outside on the windowsill. Flowering plants and the like are good camouflage, and if you tie bird food to the bottom two radials, you get brownie points from local wildlife and neighbors: songbirds are cool.
Telescopic antennas - about 10 dollars
If you choose BNC connectors, you will need a BNC-SMA and SMA-MCX adapters. With old tv/radio antennas you simply connect the center wire to the antenna by wrapping around it.
Performance is equal to a wire of a same length, in radio terms it is called a monopole.
You may add two radials similarly to the Peace antenna for performance improvements; or buy three telescopic antennas and construct an adjustable version.
A lightweight Travel Kit is a telescopic antenna with adaptors and an RTL stick with USB cord. Note on the above image Clamp-On ferrite and USB extension cord shield mod - do them, read the noise suppression post for more info.
Rubber duckySmall antennas usually used on handheld transceivers and walkie-talkies are a poor choice for beginners; they are designed to work with handhelds, where built-in electronics optimize performance. However, they work well for airband, for signals 400 MHz and up, and usually provide adequate performance if you're close to the signal source.
For general "all-in" use they are simply too small.
Most monitoring enthusiasts have one; it is like having an antenna for every frequency between 25 and 1300 MHz. Drawbacks are: expensive (50 dollar and up), look very conspicuous and nearly impossible to camouflage. Noise pickup is significant, too.
Personally, I use the 80 USD Scanmaster mentioned above with great results. For this post I used the 2-foot Peace antenna for a few days and didn't miss my usual signals.
Before you part with your hard-earned money buy a Low-Noise Amplifier (LNA) to be used with your homemade or telescopic antenna - so you can wring out the last ounce of performance from your existing equipment.
An LNA costs 25 USD, see review on this blog.
Tips and tricks
Frequencies in a country are public: Called "frequency allocations" or "frequency band plans", they are an excellent source of information. Google the terms above and you'll get a government document detailing who uses what frequencies. USA frequency allocation chart here, UK here.
Police frequencies: I know you want them. Bad news: most law enforcement / government communications went digital few years ago (you can't hear them), and sorry, I have no experience nor interest in them anyway.
Frequencies are around 120-130 MHz, for exact frequencies google "your local airport's name" and "frequencies". Normally four frequencies are given: Approach, Tower, Ground and ATIS.
Approach deals with planes approaching the Airport, Tower handles landing, and as soon wheels touch the ground Ground directs the bird to terminals. Take-off is in reverse. Larger international airports might have more than one frequencies, e.g. Heathrow has four Approach.
ATIS is continuous weather and landing info for planes, useful for weather check in the morning.
Planes are IMHO the best to listen to.
Where is what / Popular frequencies:
The following is just a quick recap for a total newbie. Really, spend about ten minutes with a frequency allocation chart - even if you're a seasoned veteran.
Mode (top, left of screen) is in brackets.
Broadcast radio (WFM): between 90-108 - signals are strong, everything receives this.
Airplanes talk (AM): around 120-130 MHz. - signals are strong, you need a dedicated antenna to hear airplanes far away. The Peace antenna above is ideal for airplanes.
Weather satellites (WFM): 137 MHz - weak signal, special antenna.
Marine channels and AIS (NFM): 162 MHz.
Taxis and speech (NFM): 160 MHz - an AIS antenna is just as good.
Shopping centre / mall radios (NFM): 440-480 MHz - also radio amateurs.
Airplane positions signals a.k.a ADS-B (AM): 1090 MHz.