Zagreb city center is divided in two - the Down town and the Upper town. In the Down town you should see (and do) Botanical garden, Green horseshue, Ban Jelačić square (the central one), Flower square (the nice one, ideal for coffee drinking and lazing on the sun), Tkalčićeva street (filled with little cafes and clubs, charming).

Down town and upper town are connected with Radićeva and Mesićeva street, as well as with the funicular, that is in Ilica street about 100m from Ban Jelačić square.
On Upper town and Kaptol you should see (and do) the Cathedral, St. Marco's church, Lotrscak tower (great belvedere and a canon that blows every noon), and many little streets up there.

If you are more for nature I can recommend you to visit the Maksimir park; it's a big monument of nature with cultivated and wild parts, and five lakes. Very nice in spring and summer! Also, Sljeme mountain just next to Zagreb can be reached by trams 14 and 15. There you should visit Medvedgrad, the old fortress. Best way to reach it is to go on tram 14 to the last station in direction Mihaljevac, and than take the bus in direction Šestine for few stations, and just go ahead towards the mountains, follow the signs... .

Zagreb is mostly flat and with wide streets, so if you don't have a car, and you do not want to rumple in stuffy trams the best way to see Zagreb is either by walk or by bicycle, just as many people do!