You can explore Madrid by public transport for next to nothing, traveling as far as you like on the urban red line buses for 1€ a trip. Even more economical is the 10-pack Metrobus ticket (9€), valid for both buses and the metro, which works out to only .90€ a trip. (A one-way combined bus and metro ticket will otherwise cost you 2€). You can buy tickets at any metro station or newspaper kiosk. To orient yourself, pick up a Consorcio Transportes de Madrid from one of the kiosks in the Puerta del Sol (1.50€). This city bus map marks all bus routes clearly. I've chosen a few of the best red line bus trips around the city. Hope you enjoy them.
33: Príncipe Pío to Casa de Campo -- This short run starts at the Príncipe Pío bus, metro, and train junction at the end of Paseo de la Florida, just northwest of the Royal Palace. First you follow the tree-lined avenue to the west of the Campo del Moro, and then you turn right across the Manzanares River at Puente de Toledo (Toledo Bridge) before passing through the built-up zone of Puerta del Angel. From here you cross the busy Paseo de Extremadura to enter the green parklands of the Casa de Campo, stopping at the Parque de Attracciones (a fun park for teenagers), Batán (where bulls are penned in a small farm building prior to participating in corridas at the Ventas bullring), and the zoo. It's a great trip for families with children. Service runs every 15 minutes from 10am to 10pm.
54: Atocha to Vallecas -- This route starts at Atocha and heads southeast along the Ciudad de Barcelona and Albufera avenues, via the lively junction of Puente Vallecas and across the M-40 highway to the Villa de Vallecas -- a self-contained town which retains its earthy individuality (it still has its own fútbol team) in spite of now being part of the Madrid community. "Gentrification" in the form of duplex flats, pedestrian paseos, and tree-lined avenues has transformed the center, but it remains at heart a traditional character-filled place, whose huge, 18th-century church of San Pedro Ad Vincula is a cultural monument. On your way back to Madrid, jump off at the Calle Pío Felipe stop just above the Buenos Aires metro station and stroll on to the high grassy knolls of Cerro del Tío Pío park. From here you can enjoy panoramic views of Madrid, and on very clear days you can see the Gredos Mountains of Avila province 100km (60 miles) to the west. Service runs every 5 minutes from 6am to 10pm.
75: Callao to Colonia Manzanares -- This route takes you up the Gran Vía to Plaza España, where you turn left down the Cuesta de San Vicente to the Príncipe Pío junction, then west along Avenida de la Florida to turn again left opposite the Casa Mingo restaurant and the Ermita de San Antonio de la Florida (an ornate chapel that contains some important Goya frescoes). Crossing the bridge over the Manzanares River, you enter the uniquely laid-back world of Colonia de Manzanares, which is nestled between the river and Casa de Campo. Its center is a small conglomeration of shops and cafes bordered by avenues of mature residential villas with gardens; on its riverside promenade you may see people fishing. The unexpected aura of peace and relaxation of this charming backwater seems light years -- rather than just a 20-minute bus ride -- away from the bustling Gran Vía. Service runs every 10 to 15 minutes from 6:15am to 11:30pm.
146: Callao to Barrio de la Concepción -- Here is your chance to check out the changing architectural styles and moods as you progress along the length of Madrid's great east-west artery, Calle Alcalá. Starting at the center of the Gran Vía, you first pass the emblematic Cibeles fountain-statue and flamboyant turn-of-the-20th-century Palacio de Comunicaciones (Post Office) before continuing uphill to the neoclassical Puerta de Alcalá archway, opposite the main entrance to Retiro Park. From here the avenue extends farther east with the Retiro and Neo-Mudéjar Antiguas Escuelas Aguirre building on your right, plus the shop-filled Velázquez and Príncipe Vergara streets of the elegant 19th-century Salamanca district on your left. Passing through Goya and Manuel Becerra squares to the Ventas bullring, you finally cross the M-30 bridge and turn left into the newer (ca. 1940s), homelier district of Barrio de la Concepción, which runs parallel to Alcalá. Step off at the tiny Calero park and stroll among the flowers, trees, and playgrounds, where hordes of toddlers play on swings next to a pine-shrouded open-air summer cinema -- one of only two in Madrid. Then relax at an outdoor terrace table in the adjoining promenade-like Calle Virgen de Nuria and enjoy a Mahou beer and tapas in the sun. Service runs every 5 to 10 minutes from 7am to 11:30pm.