If it's surfing lessons you want, you'd do better to look somewhere up in Praia de Caparica, which is over the Tagus from Lisbon - in good traffic if you're driving it's about 20 minutes from when you get on the Ponte 25 de Abril (big red suspension bridge!). In bad traffic you'll hit the queues for the coast (it's where the Lisboetas go for the day in August) a few mins off the bridge. And driving back to town late afternoon/evening is always longer! But for all that, it's worth it: I grew up spending my summers at Caparica and though it's larger now, it's an authentic Portuguese seaside town with few crappy touristic traps. Plenty of good seafood places to eat at varying prices, more artisanal ice cream places than you can shake a stick at, and the miles and miles of beaches are incomparable. But it's often rough surf - hence, great for surfing and paragliding - less so for casual swimming unless it's calm. Look for the lifesaver stations and swim close to them, and check the flags denoting safe swimming. Loads of surf schools along that coast, google them. Algarve is not really a surfing place though you'll find more surf schools the further west you go (and the water will be appreciably colder!).
In the city itself, I like to start at the Castelo do São Jorge and enjoy the views, then walk downhill, getting gently lost in the various neighbourhoods until you end up in the vicinity of the Praça do Comercio (Black Horse Square) which is great for photos but don't get a taxi from here! They are crooks, and the one we stupidly hired last year nearly killed us, getting back to Belém. If you can't walk far, then you could hire a tuk-tuk and ask to drive not too fast, and see the sights on the way down. On foot, you could go into the Sé, the medieval gothic cathedral which is wonderful and has Roman remains visible from walkways in the cloisters.
There are trams you can take - look up the route maps online - the famous 28 route is well worth getting on but will be rammed with tourists in summer and also with pickpockets. Other tram routes are available.
Elsewhere in the city, the Tile Museum in the east (Museu de Azulejos) is gorgeous and quiet, if you find you love the blue and white tile murals you'll see everywhere. There's a nice cafe in the courtyard too. Further east still is the Expo site which is pretty spectacular to explore, has some nice places to eat, and also the world-class aquarium which is fantastic for any age.
If you like art, the under-visited Museu Calouste Gulbenkian is a stunner, in a modernist building set in beautiful gardens and stuffed full of art works from all periods (it's all from a private collection from a super-rich shipping magnate...). I saw John Malkovitch there one time, which was pretty cool (he lives in town sometimes).
West of the city centre, Belém (which isn't, strictly speaking, part of Lisbon proper) is a beautiful place to walk around (though you will have to navigate from the river front to the buildings just behind, via footbridges over the rail tracks and littoral road - it's a bit of a walk at times!). There are lovely gardens to walk in and it's easy to get into 'normal Lisbon' just by walking five minutes away from the major attractions. The river frontage is beautiful and very photogenic, taking you past the cute miniature (but functional) Torre de Belém, monument to the Explorers, a tiny brick lighthouse and other landmarks.
A couple of hundred metres inland (but over those pesky rail tracks) are all the wonderful places of Belém, including the medieval monastery of Jeronimós, the church attached to it, the archaeological and maritime museums, planetarium, and the famous Pastéis de Belém (which will be a bear garden in peak summer, so go very early or later, and remember that the famously grumpy, surly waiters have thousands of tourists who don't speak Portuguese through the doors every day 
). Failing that, if you don't fancy queuing in hot sunshine for 45 minutes in order to down an espresso and a pastel, any other cafe will do killer pastéis de nata: we like 'A Padaria Portugesa (I think of it as the Lisbon Greggs! It's not, it's much more upmarket) but the stand-up cafe at the front of the Pingu Doce supermarkets will do you a pastel and small coffee for under 1 euro 50 and they're pretty nice... The cultural centre, just in front of Jeronimós, is really nice for a wander around (and shade!) and has lovely shops and cafes - it also used to have a really good modern art museum which has just closed but hopefully something similar will be open there now. Also concerts, often open-air, in the evening.
If you want to get out of the centre, get a taxi out to a palace or two - Fronteira and Queluz are both beautiful and worth a visit. They're really pretty close to the centre of the city but not a walk away. Further off still (look into taking the train!) is Sintra and all the (busy, crowded) palaces and gardens you could ever want. Sintra is pretty unique but very very busy in peak season. If you hire a car, I'd still be tempted to take the train instead! There are some stunning beaches on that stretch of the coast beyond Sintra, and they're quiet. It can be quite foggy and windy up there - bit of a microclimate.
In the Algarve, Carvoeiro is a bit built-up now but there are stunning beaches all along that stretch. As a rule if you want a bit more peace on the beach, look at the map and select a long broad beach, drive there (parking will be better for long beaches as there'll be more car parks!) and then walk along until you get to the level of people you're happy with!
Somewhere quite undervisited are the island beaches in the east of the Algarve, accessible by boat (little ferries or water taxis) from places like Cabanas, near Tavira. The boats will take you to sandbar islands (they're quite big, have the odd restaurant and bar there but take some food, in case) which are a very different experience from the bustle of most Algarve resorts. Tavira is a beautiful town with loads of very pretty churches.
Inland Algarve is undervisited but lovely too, and very peaceful (when the mountains aren't on fire). Monchique and other mountain towns nearby are worth a drive up. Take a cardi! Silves, further west, is famously beautiful and busy in summer, but fun to walk round the medieval Moorish streets - there's usually a 'Moorish' street fair there in August, and the castle at the top is worth a visit.
Wherever you go, keep an eye out for proper ice cream (gelataria) places. Nothing like them anywhere else (well maybe Italy)! The best ones will use fresh seasonal fruit. You can't get ice cream or sorbet made with fresh melon, passionfruit, peach, pear, pineapple, etc that readily in the UK. I don't know how much we spend on gelados in Portugal every year but it's a significant part of our holiday budget! Santini in Lisbon and along that coastline is famous but pretty pricy - there are other places. Gelato Daverro is really good, so is Artisani (££s) and Nanarellas. In the Algarve there will be loads of independent mom-and-pop run gelaterias and they're always worth a visit.
This is my country and my city: I'd happily live there if circumstances allowed.