What type of derailleur is that? If it is not a Shimano derailleur from ~1988 or later then the shifter and derailleur are incompatible. It is hard to tell from the picture, but it does not look like a modern Shimano indexed shifting derailleur.

Also, make sure it is a hyperglide compatible modern freewheel or cassette - look closely at the teeth on the rear gear cluster. When viewed from the side, the gears should have 'features' built in like little ridges and slightly differently shaped teeth. If all the teeth are sthe same shape and square then the rear cluseter is also incompatible.

One last thing is that the shift cable housing really should be 'compressionless' shift cable housing - the type where the wires that give the housing its stiffness, right under the plastic, should run lengthwise, not spiral wound.

If someone simply slapped STI shifters onto an old bike then these things would generally not be compatible unless they were changes also.

Luckily, a new derailleur that will work with that shifter and with a 7 speed hyperglide gear cluster costs as little as $15.
Or you can find a set of friction bar-end shifters and not change the derailleurs. You should also replace the brake levers if you go this route.

Also, the front derailleur should be a modern Shimano road derailleur to work properly, but as I recall Sora front shifting is always tough to get perfect. Bar end shifters and new brake levers are the best way to fix this.

Other things may be worn out or need replacing but this is all I can suggest based on the information you provided.