Many Lavaux villages are reachable by regional trains from Lausanne: Lutry, Villette, Cully, Epesses, Rivaz and St-Saphorin with the S1 and S3 trains (the S3 doesn't stop everywhere); Grandvaux with the S2 and Puidoux-Chexbres with the S2 and S21. There is also S31 train connecting Vevey with Puidoux-Chexbres. All trains run hourly.
Lausanne city bus 9 brings you to the west corner of Lavaux area in Lutry. Bus 66 (runs irregularly on week days) brings you directly from the Lausanne's center to Grandvaux and all places above the vineyards. Alternatively, take city bus 7 to the terminus "Val-Vert" and change to bus 47. Check the schedule in advance, the connection time could be very long. Buses 65 and 75 (start from the Lausanne Metro stop "La Sallaz") bring you to the Lavaux's high altitude communes of Savigny and Forel. These two villages make ideal starting points for countryside walks, or family cycle trips on quiet lanes. (cycle transport is permitted on the buses, and for the adventurous but lazy, electric cycles can be rented in Forel).
One of the nicest ways to spend an afternoon anywhere is to take a boat from the port of Ouchy on the Lakefront of Lausanne or from Vevey or Montreux. Enjoy the terraces which produced it roll by at a leisurely steam-powered pace. It's possible (although a much longer ride) to take the same steamboat all the way from Geneva or Evian. The steamboat ports in the Lavaux include Lutry, Cully, Rivaz, and Vevey.
The regional trains stop at many of the villages along the way and can be used to get around . However, the Lausanne-Montreux line runs close to the bottom of the Lavaux, near the big lake, so you are guaranteed to see almost nothing on the train ride (except gorgeous views of Lac Léman and the Savoie Alps between Lutry and Vevey), and the train stations are often in the suburban commerce centers of the larger municipalities, away from the vineyards themselves. There is also a Train-des-Vignes that runs from Vevey to Puidoux which can be quite scenic if you aren't fit enough to hike or walk around the Lavaux. There is also a funicular from Vevey to Mont-Pelerin station (where you can walk in perhaps 1.5 hours to the Mont-Pelerin summit at 1080 m which has a viewing tower), a train from Vevey to Les Pleiades (1360 m), and several others. Although not exactly the Lavaux, the cogwheel train from Montreux to Rochers-de-Naye (2042 m) is great because of excellent views on a clear day (although slightly expensive).
A fit traveler should be able to reach the village of Cully, in the center of the Lavaux from the port of Ouchy in Lausanne in about an two hours. If you're feeling more ambitious consider the popular walking tour of Lavaux vineyards. The whole walk from Chateau d'Ouchy (Lausanne) to Chateau de Chillion (Montreux) takes about 9 hours, not counting stops for wine-tasting and picnics. This should make for a relaxing 3-day itinerary if you book hotels at villages along the way. The Montreux-Vevey tourism board offers an excellent brochure with a detailed map of the entire bas-lavaux region called "A la découverte des terraces de Lavaux". It can be obtained upon arrival in Lausanne at the tourist office immediately outside the train station or downloaded from their website .
For no apparently good reason, it is much harder to find official guides detailing the footpaths and bike trails of Savigny and Forel, but the region is packed with gems for the discerning explorer, and is liberally peppered with charming, inexpensive places to stay. At least one web-site  provides an excellent source of attractive bed and breakfast accommodations throughout the entire Lavaux (haut et bas), which are not to be found through the Montreux-Vevey tourist board's site .
Cyclists with strong legs and a decent bicycle may find the Lavaux to be a great place to cycle some hills, and the entire Lausanne-Montreux stretch is easily doable in anywhere from under 2 hours (if you take the uninteresting path the waterfront level) or in a full day (if you take the scenic and paved Route de la Corniche along the vineyards and make plenty of photo stops along the way). Be prepared to walk your bicycle on some roads that are too steep, test your brakes very well, and watch out for extremely steep sections of roads around the corner when descending along vineyard routes. Cycles could be rented in Lausanne for those without one in Switzerland. Get the brochure described above if you don't have a cycling map. For the more adventurous, why not also discover the Lavaux's hinterlands by pedalpower. An abundance of quiet well signed cyclable routes cover the unspoiled countryside between the Mont Pelerin (above Chardonne) and the communes of Puidoux, Forel and Savigny, from where you can freewheel all the way back down to the lakeshore at Lutry (if your bike has reliable breaks).
If you are athletic, cycling is probably the best way to daytrip the bas-Lavaux because you will always be going right alongside the vineyards and villages rather than at the train stations below, and unlike driving, you can stop anywhere at ease to enjoy the breathtaking scenery. Start in Lausanne's waterfront at Ouchy, and simply hug the coast until you get to Lutry's old town; turn left, go uphill, and look for the Chateau de Bertholod sitting amongst vineyards. From there, simply follow a map, signs, or your desires eastward; it's hard to not reach Montreux because you will always have the giant lake in view to guide you. The Route de la Petite Corniche which later becomes the Route de la Corniche are particularly scenic. You can even stop in Chexbres for a budget lunch at the Coop, and then continue onwards. When you reach Montreux, catch an S train back to Lausanne with a bicycle ticket, or cycle back via the flat and fast roads at waterfront level.
The Lavaux could make for some nice drives, but this is discouraged. You will miss the best views and the peacefulness of the place if you stay in your car for too long.
One of the great joys of spending your leisure time in a place like the Lavaux can be exploring on foot. There are thousands of footpaths and stairways linking the vineyards with the major villages and the cities of Lausanne and Vevey. For the fit, a trip up through the vineyards to the Tour de Gourze  in the commune of Riez will reward you with fantastic views of the entire length of the Lac Léman as well as the possibility of a tempting fondue on an atmospheric terrace known only to the locals. The Tour can be reached by car except in winter months, but beware, the road is narrow and steep.
There is a vineyard called Fischer Vins from a walking distance to Cully station. There is an outlet for locally grown wine where you can enjoy the stunning view and taste 3 wines for 12 CHF (as of September 2018). More wines can be had on the spot or to take away.
This site  provides details of a good selection of attractive, affordable B&Bs throughout the Lavaux region.
(http://www.aubergegrandvaux.ch) A nice hotel with a wiev on the lake and with a family ambiance.A Good restaurant with typical specialities and a panoramic terrasse.