Things to do on a sunny day

Download an Audio Tour and enjoy a self-guided walking tour taking in all the main central sights, or buy the Walk Frankfurt book, paperback and kindle formats, available on global Amazon sites. This blog page highlights all the main Frankfurt attractions which are typical during a Walk Frankfurt tour.
Simply relax along the River Main and watch the World pass by. On the south bank of the river, during the summer months, boats are moored by the Eiserner Steg where you can buy a coffee, beer or whatever you fancy. By the Untermainbrücke there is also the Maincafe, right on the river bank, with lots of outdoor seating.
Join in with local life at the farmer's markets. On Thursday and Saturday the market is hosted at the Konstablerwache. On Friday there is a market on Schillerstraße (09:00 - 18:30), right outside the Stock Exchange (Börse).
Take a 100 minute Primus Line cruise up and down the Main river. The boats offer refreshment on board and a cool breeze on a hot day.
Go for a walk around the Frankfurter Wallanlage. The ring of parkland that surrounds the city centre follows the old path of the 14th century city battlements and is approximately 5km long with interesting art and artefacts to see on the way.
Walk east along the northern banks of the River Main towards the European Central Bank. There you will find a cafe/restaurant Oosten, with great food and a sunny terrace.
Enjoy cafe life in Frankfurt city centre. This link suggests a few places to hang out and look like a real local.

Maincafe, near to the Untermainbrücke

Heron enjoying the lake in the Wallanlage

Primus Line kiosk, by the Eiserner Steg

The ECB building where you will find cafe Oosten

Things to do on a rainy day

Visit the multiple museums in Frankfurt - if you are going to use the public transport consider the Frankfurt Card which gives you free travel and 20% discount on entry to many museums and attractions. If you don't need transport then consider buying a Museums Card. Once bought it will give you free entry into all of Frankfurt's museums for two days. For more information on museums pricing and buying tickets see this blog post on Frankfurt's museums.
Visit the churches - the Kaiserdom, Alte Nikolai and Katherinenkirche are all very centrally located and welcoming. Paulskirche hosts a permanent, free, exhibition dedicated to the first National Assembly of 1848 and has an amazing mural by Johannes Grützke.
Head to the Kleinmarkthalle, browse the stalls including the famous Schreiber stall serving freshly boiled sausage, snack on regional produce, or go upstairs to the first floor and sample a glass of local Riesling at the Rollanderhof.
Go and see the 16th century fresco in the Karmelitekloster. Entry is free and you can hire an English audio guide for 2€ from the reception desk with 90 minutes of information about the fresco and the artist, Jörg Ratgeb.
Take a look around a German concept shop called Manufactum, opposite the Alte Oper on Reuterweg. The shop stocks a variety of different objects and has a tasty cafe too.