Dealing with lead paint in an older home can be a tricky situation. Fortunately, however, there are several options available that can allow you to safely remove or encapsulate the paint.
The cheapest option for dealing with lead paint is to encapsulate it by painting over it with a special paint that is specifically designed to contain the lead. As long as the new paint doesn't chip, the lead will be safely contained and you don't have to worry about contamination. On corners that protrude into the room or any other areas that are prone to chips, you should consider first stripping away the paint, and then painting over it with the encapsulating paint. Again, however, this extra step is only necessary in areas that are likely to chip.
Another relatively affordable option is called enclosure. With this method, the affected area is simply covered over with new building materials. For instance, if your wall has been painted with lead based paint, you can put new sheetrock directly over the top of the old wall. Likewise, window sills can be covered with cladding made out of aluminum or vinyl. The only downside to this method is that if you ever decide to remove the enclosing material, you will still have to deal with the lead paint. 
This is one of the most labor-intensive options, and needs to be handled with extreme care. Typically lead-based paint is removed through a process of wet sanding or hand scraping. A heat gun and scraper can also work well to remove the paint. The main goal is to minimize the amount of dust that is released during the removal process. Ideally, the work should be left to a professional rather than trying to do it yourself. They have the proper tools and safety gear to avoid contamination. Here are some additional safety tips to keep in mind during the removal process: 
1. Keep children and pregnant women away from the area while work is being done. If necessary, set up temporary living arrangements off-site.
2. Only take on one room at a time and make sure that it is totally sealed off from the rest of the house.
3. Remove absolutely everything from the room that can be taken out. For anything that has to stay in the room, cover it with at least two layers of plastic. If you have carpeting, cover it with plastic that has been duct taped down to prevent dust from getting underneath.
4. Protect yourself with the proper safety gear. This includes goggles, a respirator that has been approved for lead-based paint removal, disposable coveralls, protection for your hair and covers for your shoes.
5. Avoid eating, drinking or smoking on the job.
6. Clean up thoroughly each time that you leave the work area. Throw away your coveralls and use a HEPA filtered vacuum to clean dust off of your clothes. Take a shower immediately to prevent spreading dust around.
For the removal process itself, you can choose one of the following options:
- Use paint strippers or non-flammable solvents in conjunction with a wire brush or scraper. Be sure to protect your skin from the solvent and to wear a respirator throughout the process.
- Wet sand the paint away using an electric sander with a built-in HPEA filter attachment. NEVER dry sand the surface.
- Use a heat gun with a temperature below 1100 degrees. Follow this up by hand scraping. Note: this technique is best left to the professionals since heat guns can make lead dust and vapors. Not only that, but they can also pose a threat in terms of being a fire hazard.
Methods that should NEVER be used to remove lead paint include: 
- Dry sanding or grinding
- Wet sanding without the use of a HEPA attachment
- Open-flame burning or torching
- Sand blasting or other abrasive removal methods
- Power washing - particularly if there is no way to capture the paint chips.
This video takes an in-depth look at how to deal with lead-based paint in older homes: