First of all, hunters have guns and shoot moose and bears. If you look like a moose and walk in the forest, your mother is right.
Otherwise, you can walk, looking like an American tourist, on city streets in complete safety. Well, i understand your mother's concern. my 11 yr old son popped over to Thailand two days ago from Russia (with my wife's sister), and of course i have worries, but i know tourists from Russia go to Thailand all the time. such that the Thai speak Russian!
i have been in Russia since 1991. not a year has passed that i was not here. no one gets shot. well, in Yeltsyn's mafia years, there was a much greater chance. but it is really peaceful and calm. i have been in Moscow and St Petersburg at night and walked about looking for open grocers (for food and beer!) for late night eating. even after the metro closes, i have found myself walking many blocks. i have been in medium-sized cities, too. small cities, towns and even villages. people always walk. you meet hundreds or dozens, depending on time and temperature. today was 26 below. i went about 10 blocks after dark (5pm) food shopping. i met 20-30 people is all. normally 100-150 would be out at that time. on normal cold days, 15 degrees, after 10 pm, i see a few dozen walking. some are children, 10-15 or older. always walking, as shops, libraries, schools, etc are all within 5-10 blocks. Russia is a pedestrian country. Do not go to Ukraine...YET. it used to be nice.
You must see Red Square, St Basils Cathedral, the newly built Christ The Savior Cathedral, as it is the world's tallest Orthodox Church. (if you are athiest, it is still something you should see). Go to Bolshoi Theatre or ANY live shows. Pushkin Museum, Planetarium. newly repaired Palace of People's Achievements. i hope it is nice, Putin just rebuilt it. ("some of the USSR he is bringing back") i was there in '95; Sputnik and Soyuz were pushed aside. it was a big trade hall, like a mall with no walls between stores. i liked the fountains outside better. and whatever you like, antique autos or cat performances, you should find it in Moscow. RESEARCH it well on the internet (or ask me) before you go.
Of course, if money is not a problem, there is the Trans-Siberian Railway. The Altai mountains, Lake Baikal and Irkutsk. Flying to Magadan would put you in a city which few visit. there is no road there, last i saw, maybe a trail for SERIOUS 4x4s. so it would be very remote. Don't be afraid of 4 or 5 days on a train. Russian Railroads will not put you on a wooden behch or in a bucket seat. you get a bed. rest, planning, contemplation- fun travel. i am dying to embark on a 4 or so day trip to Ulan Bator. my son wants China. so i may not get Mongolia. BUT when i go about Russia, i fear never for my personal safety. Russians are very kind, especially to foreigners.
You cannot travel internationally until you are 18. Not only will you have trouble finding accommodation, in any country, you will not be granted a Russian visa if you are still a minor. So, don't worry about getting shot because you will not be travelling just yet.
However, when you do travel to that area, it is highly unlikely that you will be shot. Nearly 1 million tourists visit Russia each year. And, except for unforeseen accidents, all return home safely.
I suggest starting with Sankt-Peterburg. It's a megapolis, but stil a touristy city.
Your issue #1 would be to get a visa. You can apply only when you are 18 - as it was mentioned before - and the procedure can be rather time-consuming and tedious.
And no, your chances to get shot are smaller than in the US. At least in Russia. Can't say anything about Ukraine.
Thereis no conflict in Russia. It is between eastern Ukraine and the Kiev government. So eastern Ukraine may be unsafe. There are also demonstrations in Kiev and western Ukraine.
depends on your interests. Culture, nature, etc.... For culture start from Moscow and Saint Petersburg. For nature - from Baikal lake.